12 Surface Area and Volume

Transcription

1 12 Surface Area and Volume 12.1 Three-Dimensional Figures 12.2 Surface Areas of Prisms and Cylinders 12.3 Surface Areas of Pyramids and Cones 12.4 Volumes of Prisms and Cylinders 12.5 Volumes of Pyramids and Cones 12.6 Surface Areas and Volumes of Spheres 12.7 Spherical Geometry Earth (p. 692) Tennis alls (p. 685) Khafre's Pyramid (p. 674) SEE the ig Idea Great lue Hole (p. 669) Traffic fic Cone (p. 656) 6) Mathematical Thinking: Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

2 Maintaining Mathematical Proficiency Finding the Area of a Circle (7.9.) Example 1 Find the area of the circle. A = πr 2 Formula for area of a circle = π 8 2 Substitute 8 for r. = 64π Simplify Use a calculator. The area is about square inches. 8 in. Find the area of the circle ft 6 m 20 cm Finding the Area of a Composite Figure (7.9.C) Example 2 Find the area of the composite figure. The composite figure is made up of a rectangle, a triangle, and a semicircle. Find the area of each figure. 16 ft 17 ft 32 ft 30 ft Area of rectangle Area of triangle Area of semicircle A = w A = 1 2 bh A = πr2 2 = 32(16) = 1 2 (30)(16) = π(17)2 2 = 512 = So, the area is about = square feet. Find the area of the composite figure m 5. 6 in cm 20 m 7 m 7 m 10 in. 3 in. 5 in. 6 cm 3 3 cm 9 cm 6 cm 7. ASTRACT REASONING A circle has a radius of x inches. Write a formula for the area of the circle when the radius is multiplied by a real number a. 637

3 Mathematical Thinking Creating a Coherent Representation Core Concept Mathematically profi cient students create and use representations to organize, record, and communicate mathematical ideas. (G.1.E) Nets for Three-Dimensional Figures A net for a three-dimensional figure is a two-dimensional pattern that can be folded to form the three-dimensional figure. w lateral face base lateral face w w lateral face lateral face h w h base Drawing a Net for a Pyramid Draw a net of the pyramid. 20 in. The pyramid has a square base. Its four lateral faces are congruent isosceles triangles. 19 in. 19 in. 19 in. 19 in. 20 in. Monitoring Progress Draw a net of the three-dimensional figure. Label the dimensions ft 2. 5 m in. 2 ft 4 ft 8 m 12 m 10 in. 10 in. 638 Chapter 12 Surface Area and Volume

4 12.1 Three-Dimensional Figures TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.A Essential Question What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? A polyhedron is a solid that is bounded by polygons, called faces. edge Each vertex is a point. Each edge is a segment of a line. Each face is a portion of a plane. face vertex Analyzing a Property of Polyhedra Work with a partner. The five Platonic solids are shown below. Each of these solids has congruent regular polygons as faces. Complete the table by listing the numbers of vertices, edges, and faces of each Platonic solid. tetrahedron cube octahedron dodecahedron icosahedron Solid Vertices, V Edges, E Faces, F tetrahedron cube MAKING MATHEMATICAL ARGUMENTS To be proficient in math, you need to reason inductively about data. octahedron dodecahedron icosahedron Communicate Your Answer 2. What is the relationship between the numbers of vertices V, edges E, and faces F of a polyhedron? (Note: Swiss mathematician Leonhard Euler ( ) discovered a formula that relates these quantities.) 3. Draw three polyhedra that are different from the Platonic solids given in Exploration 1. Count the numbers of vertices, edges, and faces of each polyhedron. Then verify that the relationship you found in Question 2 is valid for each polyhedron. Section 12.1 Three-Dimensional Figures 639

5 12.1 Lesson What You Will Learn Core Vocabulary polyhedron, p. 640 face, p. 640 edge, p. 640 vertex, p. 640 cross section, p. 641 solid of revolution, p. 642 axis of revolution, p. 642 Previous solid prism pyramid cylinder cone sphere base Classify solids. Describe cross sections. Sketch and describe solids of revolution. Classifying Solids A three-dimensional figure, or solid, is bounded by flat or curved surfaces that enclose a single region of space. A polyhedron is a solid that is bounded by polygons, called faces. An edge of a polyhedron is a line segment formed by the intersection of two faces. A vertex of a polyhedron is a point where three or more edges meet. The plural of polyhedron is polyhedra or polyhedrons. Core Concept Types of Solids Polyhedra vertex Not Polyhedra edge face prism cylinder cone pyramid sphere Pentagonal prism ases are pentagons. To name a prism or a pyramid, use the shape of the base. The two bases of a prism are congruent polygons in parallel planes. For example, the bases of a pentagonal prism are pentagons. The base of a pyramid is a polygon. For example, the base of a triangular pyramid is a triangle. Triangular pyramid ase is a triangle. Classifying Solids Tell whether each solid is a polyhedron. If it is, name the polyhedron. a. b. c. a. The solid is formed by polygons, so it is a polyhedron. The two bases are congruent rectangles, so it is a rectangular prism. b. The solid is formed by polygons, so it is a polyhedron. The base is a hexagon, so it is a hexagonal pyramid. c. The cone has a curved surface, so it is not a polyhedron. 640 Chapter 12 Surface Area and Volume

6 Monitoring Progress Help in English and Spanish at igideasmath.com Tell whether the solid is a polyhedron. If it is, name the polyhedron Describing Cross Sections Imagine a plane slicing through a solid. The intersection of the plane and the solid is called a cross section. For example, three different cross sections of a cube are shown below. square rectangle triangle Describing Cross Sections Describe the shape formed by the intersection of the plane and the solid. a. b. c. d. e. f. a. The cross section is a hexagon. b. The cross section is a triangle. c. The cross section is a rectangle. d. The cross section is a circle. e. The cross section is a circle. f. The cross section is a trapezoid. Monitoring Progress Help in English and Spanish at igideasmath.com Describe the shape formed by the intersection of the plane and the solid Section 12.1 Three-Dimensional Figures 641

7 Sketching and Describing Solids of Revolution A solid of revolution is a three-dimensional figure that is formed by rotating a two-dimensional shape around an axis. The line around which the shape is rotated is called the axis of revolution. For example, when you rotate a rectangle around a line that contains one of its sides, the solid of revolution that is produced is a cylinder. Sketching and Describing Solids of Revolution Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. a. 9 b a. 9 b. 4 5 The solid is a cylinder with a height of 9 and a base radius of 4. 2 The solid is a cone with a height of 5 and a base radius of 2. Monitoring Progress Help in English and Spanish at igideasmath.com Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid Chapter 12 Surface Area and Volume

8 12.1 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. VOCAULARY A(n) is a solid that is bounded by polygons. 2. WHICH ONE DOESN T ELONG? Which solid does not belong with the other three? Explain your reasoning. Monitoring Progress and Modeling with Mathematics In Exercises 3 6, match the polyhedron with its name In Exercises 11 14, describe the cross section formed by the intersection of the plane and the solid. (See Example 2.) A. triangular prism. rectangular pyramid C. hexagonal pyramid D. pentagonal prism In Exercises 7 10, tell whether the solid is a polyhedron. If it is, name the polyhedron. (See Example 1.) In Exercises 15 18, sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid. (See Example 3.) Section 12.1 Three-Dimensional Figures 643

9 19. ERROR ANALYSIS Describe and correct the error in identifying the solid. The solid is a rectangular pyramid. 28. ATTENDING TO PRECISION The figure shows a plane intersecting a cube through four of its vertices. The edge length of the cube is 6 inches. 20. HOW DO YOU SEE IT? Is the swimming pool shown a polyhedron? If it is, name the polyhedron. If not, explain why not. a. Describe the shape formed by the cross section. b. What is the perimeter of the cross section? c. What is the area of the cross section? REASONING In Exercises 29 34, tell whether it is possible for a cross section of a cube to have the given shape. If it is, describe or sketch how the plane could intersect the cube. 29. circle 30. pentagon 31. rhombus 32. isosceles triangle 33. hexagon 34. scalene triangle In Exercises 21 26, sketch the polyhedron. 21. triangular prism 22. rectangular prism 23. pentagonal prism 24. hexagonal prism 25. square pyramid 26. pentagonal pyramid 27. MAKING AN ARGUMENT Your friend says that the polyhedron shown is a triangular prism. Your cousin says that it is a triangular pyramid. Who is correct? Explain your reasoning. 35. REASONING Sketch the composite solid produced by rotating the figure around the given axis. Then identify and describe the composite solid. a b THOUGHT PROVOKING Describe how Plato might have argued that there are precisely five Platonic Solids (see page 639). (Hint: Consider the angles that meet at a vertex.) Maintaining Mathematical Proficiency Decide whether enough information is given to prove that the triangles are congruent. If so, state the theorem you would use. (Sections 5.3, 5.5, and 5.6) 37. AD, CD 38. JLK, JLM 39. RQP, RTS A J Reviewing what you learned in previous grades and lessons Q R S D C K L M P T 644 Chapter 12 Surface Area and Volume

10 12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10. G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find the surface area of a prism or a cylinder? Recall that the surface area of a polyhedron is the sum of the areas of its faces. The lateral area of a polyhedron is the sum of the areas of its lateral faces. Finding a Formula for Surface Area APPLYING MATHEMATICS To be proficient in math, you need to analyze relationships mathematically to draw conclusions. Work with a partner. Consider the polyhedron shown. a. Identify the polyhedron. Then sketch its net so that the lateral faces form a rectangle with the same height h as the polyhedron. What types of figures make up the net? b. Write an expression that represents the perimeter P of the base of the polyhedron. Show how you can use P to write an expression that represents the lateral area L of the polyhedron. height, h a b c c. Let represent the area of a base of the polyhedron. Write a formula for the surface area S. Finding a Formula for Surface Area Work with a partner. Consider the solid shown. a. Identify the solid. Then sketch its net. What types of figures make up the net? b. Write an expression that represents the perimeter P of the base of the solid. Show how you can use P to write an expression that represents the lateral area L of the solid. radius, r height, h c. Write an expression that represents the area of a base of the solid. d. Write a formula for the surface area S. Communicate Your Answer 3. How can you find the surface area of a prism or a cylinder? 4. Consider the rectangular prism shown. a. Find the surface area of the rectangular prism by drawing its net and finding the sum of the areas of its faces. b. Find the surface area of the rectangular prism by using the formula you wrote in Exploration 1. c. Compare your answers to parts (a) and (b). What do you notice? Section 12.2 Surface Areas of Prisms and Cylinders 645

11 12.2 Lesson What You Will Learn Core Vocabulary lateral faces, p. 646 lateral edges, p. 646 surface area, p. 646 lateral area, p. 646 net, p. 646 right prism, p. 646 oblique prism, p. 646 right cylinder, p. 647 oblique cylinder, p. 647 Previous prism bases of a prism cylinder composite solid Find lateral areas and surface areas of right prisms. Find lateral areas and surface areas of right cylinders. Use surface areas of right prisms and right cylinders. Finding Lateral Areas and Surface Areas of Right Prisms Recall that a prism is a polyhedron with two congruent faces, called bases, that lie in parallel planes. The other faces, called lateral faces, are parallelograms formed by connecting the corresponding vertices of the bases. The segments connecting these vertices are lateral edges. Prisms are classified by the shapes of their bases. base base lateral edges lateral faces The surface area of a polyhedron is the sum of the areas of its faces. The lateral area of a polyhedron is the sum of the areas of its lateral faces. Imagine that you cut some edges of a polyhedron and unfold it. The two-dimensional representation of the faces is called a net. The surface area of a prism is equal to the area of its net. The height of a prism is the perpendicular distance between its bases. In a right prism, each lateral edge is perpendicular to both bases. A prism with lateral edges that are not perpendicular to the bases is an oblique prism. height height Right rectangular prism Oblique triangular prism Core Concept Lateral Area and Surface Area of a Right Prism For a right prism with base perimeter P, base apothem a, height h, and base area, the lateral area L and surface area S are as follows. Lateral area L = Ph Surface area S = 2 + L = ap + Ph P h 646 Chapter 12 Surface Area and Volume

12 Finding Lateral Area and Surface Area Find the lateral area and the surface area of the right pentagonal prism ft 6 ft Find the apothem and perimeter of a base. 9 ft a = = P = 5(7.05) = ft a 6 ft ATTENDING TO PRECISION Throughout this chapter, round lateral areas, surface areas, and volumes to the nearest hundredth, if necessary. Find the lateral area and the surface area. L = Ph ft ft = (35.25)(9) Substitute. = Multiply. S = ap + Ph = ( ) (35.25) Substitute Formula for lateral area of a right prism Formula for surface area of a right prism Use a calculator. The lateral area is square feet and the surface area is about square feet. Monitoring Progress Help in English and Spanish at igideasmath.com 1. Find the lateral area and the surface area of a right rectangular prism with a height of 7 inches, a length of 3 inches, and a width of 4 inches. height right cylinder height oblique cylinder Finding Lateral Areas and Surface Areas of Right Cylinders Recall that a cylinder is a solid with congruent circular bases that lie in parallel planes. The height of a cylinder is the perpendicular distance between its bases. The radius of a base is the radius of the cylinder. In a right cylinder, the segment joining the centers of the bases is perpendicular to the bases. In an oblique cylinder, this segment is not perpendicular to the bases. The lateral area of a cylinder is the area of its curved surface. For a right cylinder, it is equal to the product of the circumference and the height, or 2πrh. The surface area of a cylinder is equal to the sum of the lateral area and the areas of the two bases. Core Concept Lateral Area and Surface Area of a Right Cylinder For a right cylinder with radius r, r 2 πr 2 πr height h, and base area, the lateral area L and surface area S are as follows. Lateral area Surface area L = 2πrh S = 2 + L = 2πr 2 + 2πrh h r lateral area A = 2 rh π base area A = πr 2 h base area A = r 2 π Section 12.2 Surface Areas of Prisms and Cylinders 647

13 Finding Lateral Area and Surface Area Find the lateral area and the surface area of the right cylinder. 4 m Find the lateral area and the surface area. L = 2πrh Formula for lateral area of a right cylinder = 2π(4)(8) Substitute. = 64π Simplify Use a calculator. S = 2πr 2 + 2πrh Formula for surface area of a right cylinder = 2π(4) π Substitute. = 96π Simplify Use a calculator. 8 m The lateral area is 64π, or about square meters. The surface area is 96π, or about square meters. Solving a Real-Life Problem You are designing a label for the cylindrical soup can shown. The label will cover the lateral area of the can. Find the minimum amount of material needed for the label. 9 cm Find the radius of a base. r = 1 (9) = Find the lateral area. L = 2πrh = 2π(4.5)(12) Substitute. = 108π Simplify Formula for lateral area of a right cylinder Use a calculator. 12 cm You need a minimum of about square centimeters of material. Monitoring Progress Help in English and Spanish at igideasmath.com 2. Find the lateral area and the surface area of the right cylinder. 10 in. 18 in. 3. WHAT IF? In Example 3, you change the design of the can so that the diameter is 12 centimeters. Find the minimum amount of material needed for the label. 648 Chapter 12 Surface Area and Volume

14 Using Surface Areas of Right Prisms and Right Cylinders Finding the Surface Area of a Composite Solid 3 m 4 m Find the lateral area and the surface area of the composite solid. 12 m Lateral area of solid = Lateral area of cylinder + Lateral area of prism = 2πrh + Ph = 2π(6)(12) + 14(12) 6 m = 144π Surface area of solid = Lateral area of solid + 2 ( Area of a base of the cylinder Area of a base of the prism ) = 144π (πr 2 w) = 144π [π(6) 2 4(3)] = 216π The lateral area is about square meters and the surface area is about square meters. Changing Dimensions in a Solid Describe how doubling all the linear dimensions affects the surface area of the right cylinder. 2 ft efore change After change Dimensions r = 2 ft, h = 8 ft r = 4 ft, h = 16 ft 8 ft Surface area S = 2πr 2 + 2πrh = 2π(2) 2 + 2π(2)(8) = 40π ft 2 S = 2πr 2 + 2πrh = 2π(4) 2 + 2π(4)(16) = 160π ft 2 2 mm Doubling all the linear dimensions results in a surface area that is 160π 40π = 4 = 22 times the original surface area. 8 mm 10 mm 6 mm Monitoring Progress Help in English and Spanish at igideasmath.com 4. Find the lateral area and the surface area of the composite solid at the left. 5. In Example 5, describe how multiplying all the linear dimensions by 1 affects the 2 surface area of the right cylinder. Section 12.2 Surface Areas of Prisms and Cylinders 649

15 12.2 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. VOCAULARY Sketch a right triangular prism. Identify the bases, lateral faces, and lateral edges. 2. WRITING Explain how the formula S = 2 + L applies to finding the surface area of both a right prism and a right cylinder. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the surface area of the solid formed by the net in. 8 cm 13. MODELING WITH MATHEMATICS The inside of the cylindrical swimming pool shown must be covered with a vinyl liner. The liner must cover the side and bottom of the swimming pool. What is the minimum amount of vinyl needed for the liner? (See Example 3.) 24 ft 10 in. 20 cm 4 ft In Exercises 5 8, find the lateral area and the surface area of the right prism. (See Example 1.) ft 8 ft 3 ft 3 m 8 m 9.1 m 7. A regular pentagonal prism has a height of 3.5 inches and a base edge length of 2 inches. 8. A regular hexagonal prism has a height of 80 feet and a base edge length of 40 feet. In Exercises 9 12, find the lateral area and the surface area of the right cylinder. (See Example 2.) in in. 16 cm 8 cm 14. MODELING WITH MATHEMATICS The tent shown has fabric covering all four sides and the floor. What is 4 ft the minimum amount of fabric needed to 6 ft construct the tent? In Exercises 15 18, find the lateral area and the surface area of the composite solid. (See Example 4.) cm 4 cm 1 cm cm 8 cm 4 ft 6 ft 5 ft 8 ft 7 ft 4 ft 11. A right cylinder has a diameter of 24 millimeters and a height of 40 millimeters in m 7 m 12. A right cylinder has a radius of 2.5 feet and a height of 7.5 feet. 11 in. 9 m 6 m 15 m 5 in. 650 Chapter 12 Surface Area and Volume

16 19. ERROR ANALYSIS Describe and correct the error in finding the surface area of the right cylinder. 6 cm 8 cm S = 2π (6) 2 + 2π(6)(8) = 168π cm ERROR ANALYSIS Describe and correct the error in finding the surface area of the composite solid. 16 ft 7 ft 27. MATHEMATICAL CONNECTIONS A cube has a surface area of 343 square inches. Write and solve an equation to find the length of each edge of the cube. 28. MATHEMATICAL CONNECTIONS A right cylinder has a surface area of 108π square meters. The radius of the cylinder is twice its height. Write and solve an equation to find the height of the cylinder. 29. MODELING WITH MATHEMATICS A company makes two types of recycling bins, as shown. oth types of bins have an open top. Which recycling bin requires more material to make? Explain. 6 in. 20 ft 18 ft S = 2(20)(7) + 2(18)(7) + 2π (8)(7) + 2[(18)(20) + π (8) 2 ] ft 2 36 in. 36 in. In Exercises 21 24, describe how the change affects the surface area of the right prism or right cylinder. (See Example 5.) 21. doubling all the linear dimensions 17 in. 5 in. 4 in. 22. multiplying all the linear dimensions by mm 10 in. 12 in. 30. MODELING WITH MATHEMATICS You are painting a rectangular room that is 13 feet long, 9 feet wide, and 8.5 feet high. There is a window that is 2.5 feet wide and 5 feet high on one wall. On another wall, there is a door that is 4 feet wide and 7 feet high. A gallon of paint covers 350 square feet. How many gallons of paint do you need to cover the four walls with one coat of paint, not including the window and door? 23. tripling the radius 2 yd 7 yd 24 mm 24. multiplying the base edge lengths by 1 4 and the height by 4 2 m 8 m 16 m 31. ANALYZING RELATIONSHIPS Which creates a greater surface area, doubling the radius of a cylinder or doubling the height of a cylinder? Explain your reasoning. 32. MAKING AN ARGUMENT You cut a cylindrical piece of lead, forming two congruent cylindrical pieces of lead. Your friend claims the surface area of each smaller piece is exactly half the surface area of the original piece. Is your friend correct? Explain your reasoning. In Exercises 25 and 26, find the height of the right prism or right cylinder. 25. S = 1097 m S = 480 in. 2 h 8.2 m 8 in. h 15 in. 33. USING STRUCTURE The right triangular prisms shown have the same surface area. Find the height h of prism. 20 cm Prism A 24 cm 20 cm 3 cm 6 cm Prism 8 cm h Section 12.2 Surface Areas of Prisms and Cylinders 651

17 34. USING STRUCTURE The lateral surface area of a regular pentagonal prism is 360 square feet. The height of the prism is twice the length of one of the edges of the base. Find the surface area of the prism. 35. ANALYZING RELATIONSHIPS Describe how multiplying all the linear dimensions of the right rectangular prism by each given value affects the surface area of the prism. 38. THOUGHT PROVOKING You have 24 cube-shaped building blocks with edge lengths of 1 unit. What arrangement of blocks gives you a rectangular prism with the least surface area? Justify your answer. 39. USING STRUCTURE Sketch the net of the oblique rectangular prism shown. Then find the surface area. 4 ft h 8 ft 7 ft a. 2 b. 3 c. 1 2 d. n 36. HOW DO YOU SEE IT? An open gift box is shown. a. Why is the area of the net of the box larger than the minimum amount of wrapping paper needed dd to cover the closed box? b. When wrapping the box, why would you want to use more than the minimum amount of paper needed? 37. REASONING Consider a cube that is built using 27 unit cubes, as shown. a. Find the surface area of the solid formed when the red unit cubes are removed from the solid shown. b. Find the surface area of the solid formed when the blue unit cubes are removed from the solid shown. c. Explain why your answers are different in parts (a) and (b). w 15 ft 40. WRITING Use the diagram to write a formula that can be used to find the surface area S of any cylindrical ring where 0 < r 2 < r 1. r USING STRUCTURE The diagonal of a cube is a segment whose endpoints are vertices that are not on the same face. Find the surface area of a cube with a diagonal length of 8 units. 42. USING STRUCTURE A cuboctahedron has 6 square faces and 8 equilateral triangular faces, as shown. A cuboctahedron can be made by slicing off the corners of a cube. a. Sketch a net for the cuboctahedron. b. Each edge of a cuboctahedron has a length of 5 millimeters. Find its surface area. r 2 h Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the area of the regular polygon. (Section 11.3) in m 9 in. 8 cm 6 cm 652 Chapter 12 Surface Area and Volume

18 12.3 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10. G.11.C Surface Areas of Pyramids and Cones Essential Question How can you find the surface area of a pyramid or a cone? A regular pyramid has a regular polygon for a base and the segment joining the vertex and the center of the base is perpendicular to the base. In a right cone, the segment joining the vertex and the center of the base is perpendicular to the base. APPLYING MATHEMATICS To be proficient in math, you need to analyze relationships mathematically to draw conclusions. Finding a Formula for Surface Area Work with a partner. Consider the polyhedron shown. a. Identify the polyhedron. Then sketch its net. slant height, height, h What types of figures make up the net? b. Write an expression that represents the perimeter P of the base of the polyhedron. Show how you can use P to write an expression that represents the lateral area L of the polyhedron. base edge length, b c. Let represent the area of a base of the polyhedron. Write a formula for the surface area S. Finding a Formula for Surface Area Work with a partner. Consider the solid shown. a. Identify the solid. Then sketch its net. What types of figures make up the net? b. Write an expression that represents the area of the base of the solid. c. What is the arc measure of the lateral surface of the solid? What is the circumference and area of the entire circle that contains the lateral surface of the solid? Show how you can use these three measures to find the lateral area L of the solid. slant height, radius, r d. Write a formula for the surface area S. Communicate Your Answer 3. How can you find the surface area of a pyramid or cone? 4. Consider the rectangular pyramid shown. a. Find the surface area of the rectangular pyramid by drawing its net and finding the sum of the areas of its faces. 12 ft b. Find the surface area of the rectangular pyramid by using the formula you wrote in Exploration 1. c. Compare your answers to parts (a) and (b). What do you notice? 8 ft Section 12.3 Surface Areas of Pyramids and Cones 653

19 12.3 Lesson What You Will Learn Core Vocabulary vertex of a pyramid, p. 654 regular pyramid, p. 654 slant hieght of a regular pyramid, p. 654 vertex of a cone, p. 655 right cone, p. 655 oblique cone, p. 655 slant height of a right cone, p. 655 lateral surface of a cone, p. 655 Previous pyramid cone composite solid Find lateral areas and surface areas of regular pyramids. Find lateral areas and surface areas of right cones. Use surface areas of regular pyramids and right cones. Finding Lateral Areas and Surface Areas of Regular Pyramids vertex A pyramid is a polyhedron in which the base height is a polygon and the lateral faces are triangles with a common vertex, called the vertex of the pyramid. The intersection of two lateral faces is a lateral edge. The intersection of the base and a lateral face is a base edge. The height of the pyramid is the perpendicular distance base between the base and the vertex. height Regular pyramid slant height Core Concept lateral faces Pyramid A regular pyramid has a regular polygon for a base and the segment joining the vertex and the center of the base is perpendicular to the base. The lateral faces of a regular pyramid are congruent isosceles triangles. The slant height of a regular pyramid is the height of a lateral face of the regular pyramid. A nonregular pyramid does not have a slant height. Lateral Area and Surface Area of a Regular Pyramid For a regular pyramid with base perimeter P, slant height, and base area, the lateral area L and surface area S are as follows. lateral edge base edge Lateral area L = 1 2 P Surface area S = + L = P P Finding Lateral Area and Surface Area 14 ft 10 ft 5 3 ft Find the lateral area and the surface area of the regular hexagonal pyramid. The perimeter P of the base is 6 10 = 60, feet and the apothem a is 5 3 feet. The slant height of a face is 14 feet. Find the lateral area and the surface area. L = 1 P Formula for lateral area of a regular pyramid 2 = 1 (60)(14) Substitute. 2 = 420 Simplify. S = + 1 P Formula for surface area of a regular pyramid 2 = 1 2 ( 5 3 ) (60) Substitute. = Simplify Use a calculator. The lateral area is 420 square feet and the surface area is about square feet. 654 Chapter 12 Surface Area and Volume

20 4.8 m 5.5 m Monitoring Progress Help in English and Spanish at igideasmath.com 1. Find the lateral area and the surface area of the regular pentagonal pyramid. slant height base lateral surface 8 m vertex r Right cone height vertex lateral surface height Finding Lateral Areas and Surface Areas of Right Cones A cone has a circular base and a vertex that is not in the same plane as the base. The radius of the base is the radius of the cone. The height is the perpendicular distance between the vertex and the base. In a right cone, the segment joining the vertex and the center of the base is perpendicular to the base. In an oblique cone, this segment is not perpendicular to the base. The slant height of a right cone is the distance between the vertex and a point on the edge of the base. An oblique cone does not have a slant height. The lateral surface of a cone consists of all segments that connect the vertex with points on the edge of the base. Core Concept Lateral Area and Surface Area of a Right Cone For a right cone with radius r, slant height, and base area, the lateral area L and surface area S are as follows. Lateral area L = πr r base Oblique cone Surface area S = + L = πr 2 + πr r Finding Lateral Area and Surface Area Find the lateral area and the surface area of the right cone. 6 m Use the Pythagorean Theorem (Theorem 9.1) to find the slant height. 8 m = = 10 Find the lateral area and the surface area. L = πr Formula for lateral area of a right cone = π(6)(10) Substitute. = 60π Simplify S = πr 2 + πr Use a calculator. Formula for surface area of a right cone = π(6) π Substitute. = 96π Simplify Use a calculator. The lateral area is 60π, or about square meters. The surface area is 96π, about square meters. Section 12.3 Surface Areas of Pyramids and Cones 655

21 Solving a Real-Life Problem The traffic cone can be approximated by a right cone with a radius of 5.7 inches and a height of 18 inches. Find the lateral area of the traffic cone. Use the Pythagorean Theorem (Theorem 9.1) to find the slant height. = (5.7) 2 = Find the lateral area. L = πr = π(5.7) ( ) Substitute Formula for lateral area of a right cone Use a calculator. The lateral area of the traffic cone is about square inches Monitoring Progress Help in English and Spanish at igideasmath.com 2. Find the lateral area and the surface area of the right cone. 3. WHAT IF? The radius of the cone in Example 3 is 6.3 inches. Find the lateral area. 8 ft 15 ft Using Surface Areas of Regular Pyramids and Right Cones Finding the Surface Area of a Composite Solid Find the lateral area and the surface area of the composite solid. 5 cm Lateral area of solid = Lateral area of cone = πr + 2πrh + = π(3)(5) + 2π(3)(6) = 51π Lateral area of cylinder 3 cm 6 cm Surface area of solid = Lateral area of solid = 51π + πr 2 = 51π + π(3) 2 = 60π Area of a base of the cylinder 656 Chapter 12 Surface Area and Volume The lateral area is about square centimeters and the surface area is about square centimeters.

22 Changing Dimensions in a Solid Describe how the change affects the surface area of the right cone. a. multiplying the radius by m b. multiplying all the linear dimensions by 3 2 ANALYZING MATHEMATICAL RELATIONSHIPS Notice that while the surface area does not scale by a factor of 3, the lateral 2 surface area does scale by a factor or π(15)(26) π(10)(26) = 3 2. a. efore change After change Dimensions r = 10 m, = 26 m r = 15 m, = 26 m Surface area S = πr 2 + πr = π(10) 2 + π(10)(26) = 360π m 2 Multiplying the radius by 3 2 the original surface area. S = πr 2 + πr = π(15) 2 + π(15)(26) = 615π m 2 10 m 615π results in a surface area that is 360π = times b. efore change After change Dimensions r = 10 m, = 26 m r = 15 m, = 39 m Surface area S = 360π m 2 S = πr 2 + πr = π(15) 2 + π(15)(39) = 810π m 2 Multiplying all the linear dimensions by 3 results in a surface area that 2 is 810π 360π = 9 4 ( = 3 2) 2 times the original surface area. Monitoring Progress Help in English and Spanish at igideasmath.com 4. Find the lateral area and the surface area of the composite solid. 10 ft 6 ft 5 ft 5. Describe how (a) multiplying the base edge lengths by 1 and (b) multiplying all the linear 2 dimensions by 1 affects the surface area of the 2 square pyramid. 6 m 5 m Section 12.3 Surface Areas of Pyramids and Cones 657

23 12.3 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. WRITING Describe the differences between pyramids and cones. Describe their similarities. 2. DIFFERENT WORDS, SAME QUESTION Which is different? Find both answers. Find the slant height of the regular pyramid. A 5 in. Find A. Find the height of the regular pyramid. 6 in. Find the height of a lateral face of the regular pyramid. Monitoring Progress and Modeling with Mathematics In Exercises 3 6, find the lateral area and the surface area of the regular pyramid. (See Example 1.) in mm 7.2 mm 11. ERROR ANALYSIS Describe and correct the error in finding the surface area of the regular pyramid. 4 ft 5 ft S = + 12Pℓ = (24)(4) = 84 ft2 6 ft 5 in. 5. A square pyramid has a height of 21 feet and a base edge length of 40 feet. finding the surface area of the right cone. 6. A regular hexagonal pyramid has a slant height of 15 centimeters and a base edge length of 8 centimeters. In Exercises 7 10, find the lateral area and the surface area of the right cone. (See Example 2.) ERROR ANALYSIS Describe and correct the error in cm 16 in. 11 cm 10 cm 8 cm S = 𝛑r 2 + 𝛑r 2ℓ = 𝛑 (6)2 + 𝛑 (6)2(10) 6 cm = 396𝛑 cm2 13. MODELING WITH MATHEMATICS You are making cardboard party hats like the one shown. About how much cardboard 5.5 in. do you need for each hat? (See Example 3.) 8 in. 3.5 in. 9. A right cone has a radius of 9 inches and a height of 12 inches. 10. A right cone has a diameter of 11.2 feet and a height of 9.2 feet. 658 Chapter 12 Surface Area and Volume 14. MODELING WITH MATHEMATICS A candle is in the shape of a regular square pyramid with a base edge length of 16 centimeters and a height of 15 centimeters. Find the surface area of the candle.

24 In Exercises 15 18, find the lateral area and the surface area of the composite solid. (See Example 4.) yd 4 yd 8 yd in. 5 in. 5 in cm 5 mm 7 mm 5 in 7.5 in. 24. USING STRUCTURE The sector shown can be rolled 150 to form the lateral surface area of a right cone. The lateral surface area of the cone is 20 square meters. a. Use the formula for the area of a sector to find the slant height of the cone. Explain your reasoning. b. Find the radius and the height of the cone. In Exercises 25 and 26, find the missing dimensions of the regular pyramid or right cone. 25. S = 864 in S = cm 2 12 cm h 15 in. h 4 mm In Exercises 19 22, describe how the change affects the surface area of the regular pyramid or right cone. (See Example 5.) 19. doubling the radius 20. multiplying the base edge lengths by 4 5 and the slant height by in. 4 mm x 8 in. 27. WRITING Explain why a nonregular pyramid does not have a slant height. 28. WRITING Explain why an oblique cone does not have a slant height. 29. ANALYZING RELATIONSHIPS In the figure, AC = 4, A = 3, and DC = 2. 3 in. 10 mm A 21. tripling all the linear dimensions 22. multiplying all the linear dimensions by 4 3 D E 4 m 3.6 ft 2 m 2.4 ft 23. PROLEM SOLVING Refer to the regular pyramid and right cone a. Which solid has the base with the greater area? b. Which solid has the greater slant height? c. Which solid has the greater lateral area? 4 3 a. Prove AC DEC. b. Find C, DE, and EC. c. Find the surface areas of the larger cone and the smaller cone in terms of π. Compare the surface areas using a percent. 30. REASONING To make a paper drinking cup, start with a circular piece of paper that has a 3-inch radius, then follow the given steps. How does the surface area of the cup compare to the original paper circle? Find m AC. 3 in. fold C fold A C open cup Section 12.3 Surface Areas of Pyramids and Cones 659

25 31. MAKING AN ARGUMENT Your friend claims that the lateral area of a regular pyramid is always greater than the area of the base. Is your friend correct? Explain your reasoning. 32. HOW DO YOU SEE IT? Name the figure that is represented by each net. Justify your answer. a. 35. USING STRUCTURE A right cone with a radius of 4 inches and a square pyramid both have a slant height of 5 inches. oth solids have the same surface area. Find the length of a base edge of the pyramid. 36. THOUGHT PROVOKING The surface area of a regular pyramid is given by S = + 1 P. As the number 2 of lateral faces approaches infinity, what does the pyramid approach? What does approach? What does 1 P approach? What can you conclude from 2 your three answers? Explain your reasoning. b. 37. DRAWING CONCLUSIONS The net of the lateral surface of a cone is a circular sector with radius y, as shown. x y 33. REASONING In the figure, a right cone is placed in the smallest right cylinder that can fit the cone. Which solid has a greater surface area? Explain your reasoning. 34. CRITICAL THINKING A regular hexagonal pyramid with a base edge of 9 feet and a height of 12 feet is inscribed in a right cone. Find the lateral area of the cone. a. Let y = 2. Copy and complete the table. Angle measure of lateral surface, x Slant height of cone, Circumference of base of cone, C Height of cone, h b. What conjectures can you make about the dimensions of the cone as x increases? Maintaining Mathematical Proficiency Find the volume of the prism. (Skills Review Handbook) Reviewing what you learned in previous grades and lessons 7 ft 2 ft 3 ft 10 mm = 29 mm Chapter 12 Surface Area and Volume

26 What Did You Learn? Core Vocabulary polyhedron, p. 640 face, p. 640 edge, p. 640 vertex, p. 640 cross section, p. 641 solid of revolution, p. 642 axis of revolution, p. 642 lateral faces, p. 646 lateral edges, p. 646 surface area, p. 646 lateral area, p. 646 net, p. 646 right prism, p. 646 oblique prism, p. 646 right cylinder, p. 647 oblique cylinder, p. 647 vertex of a pyramid, p. 654 regular pyramid, p. 654 slant height of a regular pyramid, p. 654 vertex of a cone, p. 655 right cone, p. 655 oblique cone, p. 655 slant height of a right cone, p. 655 lateral surface of a cone, p. 655 Core Concepts Section 12.1 Types of Solids, p. 640 Cross Section of a Solid, p. 641 Solids of Revolution, p. 642 Section 12.2 Lateral Area and Surface Area of a Right Prism, p. 646 Lateral Area and Surface Area of a Right Cylinder, p. 647 Section 12.3 Lateral Area and Surface Area of a Regular Pyramid, p. 654 Lateral Area and Surface Area of a Right Cone, p. 655 Mathematical Thinking 1. In Exercises on page 644, describe the steps you took to sketch each polyhedron. 2. Sketch and label a diagram to represent the situation described in Exercise 32 on page In Exercise 13 on page 658, you need to make a new party hat using 4 times as much cardboard as you previously used for one hat. How should you change the given dimensions to create the new party hat? Explain your reasoning. Study Skills Form a Final Exam Study Group Form a study group several weeks before the final exam. The intent of this group is to review what you have already learned while continuing to learn new material. 661

27 Quiz Tell whether the solid is a polyhedron. If it is, name the polyhedron. (Section 12.1) Sketch the composite solid produced by rotating the figure around the given axis. Then identify and describe the composite solid. (Section 12.1) Find the lateral area and the surface area of the right prism or right cylinder. (Section 12.2) ft 7. 5 in. 7 in. 9 in. 7 ft 10 cm 9 m 12 m 10 m 8. Find the lateral area and the surface area of the composite solid. (Section 12.2) 12 cm 32 cm 10 cm Find the lateral area and the surface area of the regular pyramid or right cone. (Section 12.3) cm ft m 4 3 m 8 cm 16 ft 8 m 12 ft 12. You are replacing the siding and the roofing on the house shown. You have 900 square feet of siding, 500 square feet of roofing material, and 2000 square feet of tarp, in case it rains. (Section 12.3) 12 ft 18 ft 18 ft a. Do you have enough siding to replace the siding on all four sides of the house? Explain. b. Do you have enough roofing material to replace the entire roof? Explain. c. Do you have enough tarp to cover the entire house? Explain. 662 Chapter 12 Surface Area and Volume

28 12.4 Volumes of Prisms and Cylinders TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10. G.11.D Essential Question How can you find the volume of a prism or cylinder that is not a right prism or right cylinder? Recall that the volume V of a right prism or a right cylinder is equal to the product of the area of a base and the height h. right prisms right cylinder V = h Finding Volume USING PRECISE MATHEMATICAL LANGUAGE To be proficient in math, you need to communicate precisely to others. Work with a partner. Consider a stack of square papers that is in the form of a right prism. a. What is the volume of the prism? b. When you twist the stack of papers, as shown at the right, do you change the volume? Explain your reasoning. 8 in. c. Write a carefully worded conjecture that describes the conclusion you reached in part (b). d. Use your conjecture to find the volume of the twisted stack of papers. 2 in. 2 in. Finding Volume Work with a partner. Use the conjecture you wrote in Exploration 1 to find the volume of the cylinder. a. 2 in. b. 5 cm 3 in. 15 cm Communicate Your Answer 3. How can you find the volume of a prism or cylinder that is not a right prism or right cylinder? 4. In Exploration 1, would the conjecture you wrote change if the papers in each stack were not squares? Explain your reasoning. Section 12.4 Volumes of Prisms and Cylinders 663

29 12.4 Lesson What You Will Learn Core Vocabulary volume, p. 664 Cavalieri s Principle, p. 664 Previous prism cylinder composite solid Find volumes of prisms and cylinders. Use volumes of prisms and cylinders. Finding Volumes of Prisms and Cylinders The volume of a solid is the number of cubic units contained in its interior. Volume is measured in cubic units, such as cubic centimeters (cm 3 ). Cavalieri s Principle, named after onaventura Cavalieri ( ), states that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume. The prisms below have equal heights h and equal cross-sectional areas at every level. y Cavalieri s Principle, the prisms have the same volume. h Core Concept Volume of a Prism The volume V of a prism is V = h where is the area of a base and h is the height. h h Find the volume of each prism. Finding Volumes of Prisms a. 4 cm 3 cm b. 3 cm 14 cm 2 cm 5 cm 6 cm a. The area of a base is = 1 2 (3)(4) = 6 cm2 and the height is h = 2 cm. V = h Formula for volume of a prism = 6(2) Substitute. = 12 Simplify. The volume is 12 cubic centimeters. b. The area of a base is = 1 2 (3)(6 + 14) = 30 cm2 and the height is h = 5 cm. V = h = 30(5) Substitute. = 150 Simplify. Formula for volume of a prism The volume is 150 cubic centimeters. 664 Chapter 12 Surface Area and Volume

30 Consider a cylinder with height h and base radius r and a rectangular prism with the same height that has a square base with sides of length r π. h r π r π r The cylinder and the prism have the same cross-sectional area, πr 2, at every level and the same height. y Cavalieri s Principle, the prism and the cylinder have the same volume. The volume of the prism is V = h = πr 2 h, so the volume of the cylinder is also V = h = πr 2 h. Core Concept Volume of a Cylinder The volume V of a cylinder is V = h = πr 2 h where is the area of a base, h is the height, and r is the radius of a base. r h r h Finding Volumes of Cylinders Find the volume of each cylinder. a. 9 ft 6 ft b. 4 cm 7 cm a. The dimensions of the cylinder are r = 9 ft and h = 6 ft. V = πr 2 h = π(9) 2 (6) = 486π The volume is 486π, or about cubic feet. b. The dimensions of the cylinder are r = 4 cm and h = 7 cm. V = πr 2 h = π(4) 2 (7) = 112π The volume is 112π, or about cubic centimeters. Monitoring Progress Find the volume of the solid m 5 m 8 m Help in English and Spanish at igideasmath.com 2. 8 ft 14 ft Section 12.4 Volumes of Prisms and Cylinders 665

31 Using Volumes of Prisms and Cylinders Modeling with Mathematics You are building a rectangular chest. You want the length to be 6 feet, the width to be 4 feet, and the volume to be 72 cubic feet. What should the height be? V = 72 ft 3 h 6 ft 4 ft 1. Understand the Problem You know the dimensions of the base of a rectangular prism and the volume. You are asked to find the height. 2. Make a Plan Write the formula for the volume of a rectangular prism, substitute known values, and solve for the height h. 3. Solve the Problem The area of a base is = 6(4) = 24 ft 2 and the volume is V = 72 ft 3. V = h Formula for volume of a prism 72 = 24h Substitute. 3 = h Divide each side by 24. The height of the chest should be 3 feet. 4. Look ack Check your answer. V = h = 24(3) = 72 Monitoring Progress Help in English and Spanish at igideasmath.com 3. WHAT IF? In Example 3, you want the length to be 5 meters, the width to be 3 meters, and the volume to be 60 cubic meters. What should the height be? Changing Dimensions in a Solid ANALYZING MATHEMATICAL RELATIONSHIPS Notice that when all the linear dimensions are multiplied by k, the volume is multiplied by k 3. V original = h = wh V new = (k )(kw)(kh) = (k 3 ) wh Describe how doubling all the linear dimensions affects the volume of the rectangular prism. 6 ft 4 ft efore change After change Dimensions = 4 ft, w = 3 ft, h = 6 ft = 8 ft, w = 6 ft, h = 12 ft V = h V = h Volume = (4)(3)(6) = (8)(6)(12) = 72 ft 3 = 576 ft 3 3 ft = (k 3 )V original Doubling all the linear dimensions results in a volume that is = 8 = 2 3 times the original volume. 666 Chapter 12 Surface Area and Volume

32 Changing a Dimension in a Solid Describe how tripling the radius affects the volume of the cylinder. 3 cm 6 cm efore change After change Dimensions r = 3 cm, h = 6 cm r = 9 cm, h = 6 cm Volume V = πr 2 h = π(3) 2 (6) = 54π cm 3 V = πr 2 h = π(9) 2 (6) = 486π cm 3 Tripling the radius results in a volume that is 486π 54π = 9 = 32 times the original volume. Monitoring Progress Help in English and Spanish at igideasmath.com 4. In Example 4, describe how multiplying all the linear dimensions by 1 affects the 2 volume of the rectangular prism. 5. In Example 4, describe how doubling the length and width of the bases affects the volume of the rectangular prism. 6. In Example 5, describe how multiplying the height by 2 affects the volume of the 3 cylinder. 7. In Example 5, describe how multiplying all the linear dimensions by 4 affects the volume of the cylinder. Finding the Volume of a Composite Solid 0.39 ft Find the volume of the concrete block ft 0.33 ft To find the area of the base, subtract two times the area of the small rectangle from the large rectangle ft 0.66 ft 0.66 ft 3 ft 10 ft 6 ft = Area of large rectangle 2 Area of small rectangle = 1.31(0.66) 2(0.33)(0.39) = Using the formula for the volume of a prism, the volume is V = h = (0.66) The volume is about 0.40 cubic foot. Monitoring Progress 8. Find the volume of the composite solid. Help in English and Spanish at igideasmath.com Section 12.4 Volumes of Prisms and Cylinders 667

33 12.4 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. VOCAULARY In what type of units is the volume of a solid measured? 2. COMPLETE THE SENTENCE Cavalieri s Principle states that if two solids have the same and the same at every level, then they have the same. Monitoring Progress and Modeling with Mathematics In Exercises 3 6, find the volume of the prism. (See Example 1.) 12. A pentagonal prism has a height of 9 feet and each base edge is 3 feet cm 1.8 cm 2.3 cm 2 cm 4. 4 m 1.5 m 2 m In Exercises 13 18, find the missing dimension of the prism or cylinder. (See Example 3.) 13. Volume = 560 ft Volume = 2700 yd in. 10 in. u v 5 in. 14 m 7 ft 8 ft 12 yd 15 yd 15. Volume = 80 cm Volume = in. 3 6 m 11 m In Exercises 7 10, find the volume of the cylinder. (See Example 2.) 8 cm 5 cm w 2 in. x 7. 3 ft cm 17. Volume = 3000 ft Volume = m ft 9.8 cm 9.3 ft y z 15 m 9. 5 ft m 8 ft 18 m 19. ERROR ANALYSIS Describe and correct the error in finding the volume of the cylinder. 60 In Exercises 11 and 12, make a sketch of the solid and find its volume. 11. A prism has a height of 11.2 centimeters and an equilateral triangle for a base, where each base edge is 8 centimeters. 4 ft 3 ft V = 2πrh = 2π(4)(3) = 24π So, the volume of the cylinder is 24π cubic feet. 668 Chapter 12 Surface Area and Volume

34 20. OPEN-ENDED Sketch two rectangular prisms that have volumes of 100 square inches but different surface areas. Include dimensions in your sketches. In Exercises 21 26, describe how the change affects the volume of the prism or cylinder. (See Examples 4 and 5.) 21. tripling all the linear dimensions 22. multiplying all the linear dimensions by MODELING WITH MATHEMATICS The Great lue Hole is a cylindrical trench located off the coast of elize. It is approximately 1000 feet wide and 400 feet deep. About how many gallons of water does the Great lue Hole contain? (1 ft gallons) 12 m 3 in. 8 in. 16 m 4 in. 23. multiplying the 24. tripling the base and radius by 1 the height of the 2 triangular bases 8 cm 7 cm 5 ft 12 ft 12 ft 25. multiplying the 26. multiplying the height height by 1 by in. 32. COMPARING METHODS The Volume Addition Postulate states that the volume of a solid is the sum of the volumes of all its nonoverlaping parts. Use this postulate to find the volume of the block of concrete in Example 6 by subtracting the volume of each hole from the volume of the large rectangular prism. Which method do you prefer? Explain your reasoning. 33. WRITING oth of the figures shown are made up of the same number of congruent rectangles. Explain how Cavalieri s Principle can be adapted to compare the areas of these figures. 1 in. 3 m 5 m 5 m In Exercises 27 30, find the volume of the composite solid. (See Example 6.) 34. HOW DO YOU SEE IT? Each stack of memo papers contains 500 equally-sized sheets of paper. Compare their volumes. Explain your reasoning ft 10 ft in. 2 ft 3 ft 2 ft 6 ft 8 in. 11 in in. 1 ft 4 in. 4 in. 5 ft 35. PROLEM SOLVING An aquarium shaped like a rectangular prism has a length of 30 inches, a width of 10 inches, and a height of 20 inches. You fill the aquarium 3 full with water. When you submerge a 4 rock in the aquarium, the water level rises 0.25 inch. a. Find the volume of the rock. 4 ft 2 ft b. How many rocks of this size can you place in the aquarium before water spills out? Section 12.4 Volumes of Prisms and Cylinders 669

35 36. MODELING WITH MATHEMATICS Which box gives you more cereal for your money? Explain. 41. MATHEMATICAL CONNECTIONS You drill a circular hole of radius r through the base of a cylinder of radius R. Assume the hole is drilled completely through to the other base. You want the volume of the hole to be half the volume of the cylinder. Express r as a function of R. 16 in. 4 in. 10 in. 2 in. 8 in. 10 in. 42. THOUGHT PROVOKING Cavalieri s Principle states that the two solids shown below have the same volume. Do they also have the same surface area? Explain your reasoning. 37. CRITICAL THINKING A 3-inch by 5-inch index card is rotated around a horizontal line and a vertical line to produce two different solids. Which solid has a greater volume? Explain your reasoning. h 3 in. 5 in. 3 in. 5 in. 43. PROLEM SOLVING A barn is in the shape of a pentagonal prism with the dimensions shown. The volume of the barn is 9072 cubic feet. Find the dimensions of each half of the roof. 38. CRITICAL THINKING The height of cylinder X is twice the height of cylinder Y. The radius of cylinder X is half the radius of cylinder Y. Compare the volumes of cylinder X and cylinder Y. Justify your answer. 39. USING STRUCTURE Find the volume of the solid shown. The bases of the solid are sectors of circles in. 3 π 3.5 in. x ft Not drawn to scale 8 ft 8 ft 18 ft 36 ft 44. PROLEM SOLVING A wooden box is in the shape of a regular pentagonal prism. The sides, top, and bottom of the box are 1 centimeter thick. Approximate the volume of wood used to construct the box. Round your answer to the nearest tenth. 4 cm 40. ANALYZING RELATIONSHIPS How can you change the height of a cylinder so that the volume is increased by 25% but the radius remains the same? 6 cm Maintaining Mathematical Proficiency Reviewing what you learned in previous grades and lessons Find the surface area of the regular pyramid. (Section 12.3) m cm in. 2 m 8 cm 18 in in. 670 Chapter 12 Surface Area and Volume

36 12.5 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10. G.11.D Volumes of Pyramids and Cones Essential Question How can you find the volume of a pyramid or a cone? Work with a partner. The pyramid and the prism have the same height and the same square base. Finding the Volume of a Pyramid h When the pyramid is filled with sand and poured into the prism, it takes three pyramids to fill the prism. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math, you need to look closely to discern a pattern or structure. Use this information to write a formula for the volume V of a pyramid. Finding the Volume of a Cone Work with a partner. The cone and the cylinder have the same height and the same circular base. h When the cone is filled with sand and poured into the cylinder, it takes three cones to fill the cylinder. Use this information to write a formula for the volume V of a cone. Communicate Your Answer 3. How can you find the volume of a pyramid or a cone? Section 12.5 Volumes of Pyramids and Cones 671

37 12.5 Lesson What You Will Learn Core Vocabulary Previous pyramid cone composite solid N M N K L Pyramid Q P K M Pyramid R J L Find volumes of pyramids. Find volumes of cones. Use volumes of pyramids and cones. Finding Volumes of Pyramids Consider a triangular prism with parallel, congruent bases JKL and MNP. You can divide this triangular prism into three triangular pyramids. N K L P Triangular prism M J N K L Triangular pyramid 1 M K L Triangular pyramid 2 M J N P L Triangular pyramid 3 You can combine triangular pyramids 1 and 2 to form a pyramid with a base that is a parallelogram, as shown at the left. Name this pyramid Q. Similarly, you can combine triangular pyramids 1 and 3 to form pyramid R with a base that is a parallelogram. In pyramid Q, diagonal KM divides JKNM into two congruent triangles, so the bases of triangular pyramids 1 and 2 are congruent. Similarly, you can divide any cross section parallel to JKNM into two congruent triangles that are the cross sections of triangular pyramids 1 and 2. y Cavalieri s Principle, triangular pyramids 1 and 2 have the same volume. Similarly, using pyramid R, you can show that triangular pyramids 1 and 3 have the same volume. y the Transitive Property of Equality, triangular pyramids 2 and 3 have the same volume. The volume of each pyramid must be one-third the volume of the prism, or V = 1 3 h. You can generalize this formula to say that the volume of any pyramid with any base is equal to 1 the volume of a prism with the same base and height because you can divide 3 any polygon into triangles and any pyramid into triangular pyramids. Core Concept Volume of a Pyramid The volume V of a pyramid is V = 1 3 h where is the area of a base and h is the height. h h M Find the volume of the pyramid. Finding the Volume of a Pyramid 672 Chapter 12 Surface Area and Volume V = 1 h Formula for volume of a pyramid 3 = 1 3 ( ) (9) Substitute. = 36 Simplify. The volume is 36 cubic meters. 9 m 4 m 6 m

38 Finding Volumes of Cones Consider a cone with a regular polygon inscribed in the base. The pyramid with the same vertex as the cone has volume V = 1 h. As you increase the number of sides of 3 the polygon, it approaches the base of the cone and the pyramid approaches the cone. The volume approaches 1 3 πr2 h as the base area approaches πr 2. Core Concept Volume of a Cone The volume V of a cone is V = 1 3 h = 1 3 πr2 h where is the area of a base, h is the height, and r is the radius of the base. h r r h Find the volume of the cone. Finding the Volume of a Cone 4.5 cm V = 1 3 πr2 h Formula for volume of a cone = 1 3 π (2.2)2 4.5 Substitute. = 7.26π Simplify Use a calculator. 2.2 cm The volume is 7.26π, or about cubic centimeters. Monitoring Progress Find the volume of the solid. Help in English and Spanish at igideasmath.com m 20 cm 8 m 12 cm Section 12.5 Volumes of Pyramids and Cones 673

39 Using Volumes of Pyramids and Cones Using the Volume of a Pyramid Originally, Khafre s Pyramid had a height of about 144 meters and a volume of about 2,218,800 cubic meters. Find the side length of the square base. Khafre s Pyramid, Egypt V = 1 h Formula for volume of a pyramid 3 2,218, x2 (144) Substitute. 6,656, x 2 Multiply each side by 3. 46,225 x 2 Divide each side by x Find the positive square root. Originally, the side length of the square base was about 215 meters. Monitoring Progress Help in English and Spanish at igideasmath.com 3. The volume of a square pyramid is 75 cubic meters and the height is 9 meters. Find the side length of the square base. 4. Find the height of the 5. Find the radius of the cone. triangular pyramid. V = 24 m 3 13 in. h r 6 m 3 m V = 351 π in m 6 m 9 m Changing Dimensions in a Solid Describe how multiplying all the linear dimensions by 1 affects the volume of the 3 rectangular pyramid. efore change After change Dimensions = 9 m, w = 6 m, h = 18 m = 3 m, w = 2 m, h = 6 m Volume V = 1 3 h = 1 3 (9)(6)(18) = 324 m 3 V = 1 3 h = 1 3 (3)(2)(6) = 12 m 3 Multiplying all the linear dimensions by 1 results in a volume that 3 is = 1 27 ( = 3) 1 3 times the original volume. 674 Chapter 12 Surface Area and Volume

40 Changing a Dimension in a Solid ANALYZING MATHEMATICAL RELATIONSHIPS Notice that when the height is multiplied by k, the volume is also multiplied by k. V original = 1 3 πr2 h V new = 1 3 πr2 (kh) = (k) 1 3 πr2 h = (k)v original Describe how doubling the height affects the volume of the cone. efore change After change Dimensions r = 3 ft, h = 5 ft r = 3 ft, h = 10 ft Volume V = 1 3 πr2 h = 1 3 π(3)2 (5) = 15π ft 3 V = 1 3 πr2 h = 1 3 π(3)2 (10) = 30π ft 3 5 ft Doubling the height results in a volume that is 30π = 2 times the 15π original volume. 3 ft Monitoring Progress Help in English and Spanish at igideasmath.com 6. In Example 4, describe how multiplying all the linear dimensions by 4 affects the volume of the rectangular pyramid. 7. In Example 4, describe how multiplying the height by 1 affects the volume of the 2 rectangular prism. 8. In Example 5, describe how doubling the radius affects the volume of the cone. 9. In Example 5, describe how tripling all the linear dimensions affects the volume of the cone. Finding the Volume of a Composite Solid Find the volume of the composite solid. 6 m Volume of solid = Volume of cube + Volume of pyramid 6 m = s h Write formulas. 3 = (6)2 6 Substitute. 6 m 6 m 5 cm = Simplify. = 288 Add. The volume is 288 cubic meters. 10 cm 3 cm Monitoring Progress 10. Find the volume of the composite solid. Help in English and Spanish at igideasmath.com Section 12.5 Volumes of Pyramids and Cones 675

41 12.5 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. REASONING A square pyramid and a cube have the same base and height. Compare the volume of the square pyramid to the volume of the cube. 2. COMPLETE THE SENTENCE The volume of a cone with radius r and height h is 1 the volume 3 of a(n) with radius r and height h. Monitoring Progress and Modeling with Mathematics In Exercises 3 and 4, find the volume of the pyramid. (See Example 1.) 3. 7 m 12 m 16 m 4. 3 in. In Exercises 5 and 6, find the volume of the cone. (See Example 2.) mm 10 mm 6. 1 m 4 in. In Exercises 7 12, find the missing dimension of the pyramid or cone. (See Example 3.) 3 in. 7. Volume = 912 ft 3 8. Volume = 105 cm 3 19 ft 15 cm 2 m 11. Volume = 224 in Volume = 198 yd 3 12 in. 8 in. 11 yd 9 yd 13. ERROR ANALYSIS Describe and correct the error in finding the volume of the pyramid. 5 ft 6 ft V = 1 3 (6)(5) = 1 3 (30) = 10 ft ERROR ANALYSIS Describe and correct the error in finding the volume of the cone. 10 m 6 m V = 1 3 π(6)2 (10) = 120π m 3 s 7 cm w 9. Volume = 24π m Volume = 216π in OPEN-ENDED Give an example of a pyramid and a prism that have the same base and the same volume. Explain your reasoning. h r 18 in. 16. OPEN-ENDED Give an example of a cone and a cylinder that have the same base and the same volume. Explain your reasoning. 3 m 676 Chapter 12 Surface Area and Volume

42 In Exercises 17 22, describe how the change affects the volume of the pyramid or cone. (See Examples 4 and 5.) 17. doubling all the linear dimensions 7 in. 4 in. 18. multiplying all the linear dimensions by cm 12 cm 19. multiplying the base edge lengths by tripling the radius 9 ft 21. multiplying the height by 4 6 in. 11 ft 10 in. 16 cm 9 m 5 m 22. multiplying the height by cm 18 cm In Exercises 23 28, find the volume of the composite solid. (See Example 6.) 12 cm in. 12 in. 12 in m 5.1 m 5.1 m 29. REASONING A snack stand serves a small order of popcorn in a cone-shaped container and a large order of popcorn in a cylindrical container. Do not perform any calculations. 3 in. 3 in. 8 in. 8 in. $1.25 $2.50 a. How many small containers of popcorn do you have to buy to equal the amount of popcorn in a large container? Explain. b. Which container gives you more popcorn for your money? Explain. 30. HOW DO YOU SEE IT? The cube shown is formed by three pyramids, each with the same square base and the same height. How could you use this to verify the formula for the volume of a pyramid? cm 12 cm 10 cm 9 cm cm 5 cm 5 cm In Exercises 31 and 32, find the volume of the right cone ft yd cm 3 cm 3 cm ft 12 ft 9 ft 33. MODELING WITH MATHEMATICS A cat eats half a cup of food, twice per day. Will the automatic pet feeder hold enough food for 10 days? Explain your reasoning. (1 cup 14.4 in. 3 ) 2.5 in. 7.5 in. 4 in. Section 12.5 Volumes of Pyramids and Cones 677

43 34. MODELING WITH MATHEMATICS During a chemistry lab, you use a funnel to pour a solvent into a flask. The radius of the funnel is 5 centimeters and its height is 10 centimeters. You pour the solvent into the funnel at a rate of 80 milliliters per second and the solvent flows out of the funnel at a rate of 65 milliliters per second. How long will it be before the funnel overflows? (1 ml = 1 cm 3 ) 35. ANALYZING RELATIONSHIPS A cone has height h and a base with radius r. You want to change the cone so its volume is doubled. What is the new height if you change only the height? What is the new radius if you change only the radius? Explain. 36. REASONING The figure shown is a cone that has been warped but whose cross sections still have the same area as a right cone with equal radius and height. Find the volume of this solid. Explain your reasoning. 3 cm 2 cm 37. CRITICAL THINKING Find the volume of the regular pentagonal pyramid. Round your answer to the nearest hundredth. In the diagram, m AC = MAKING AN ARGUMENT In the figure, the two cylinders are congruent. The combined height of the two smaller cones equals the height of the larger cone. Your friend claims that this means the total volume of the two smaller cones is equal to the volume of the larger cone. Is your friend correct? Justify your answer. 40. MODELING WITH MATHEMATICS Nautical deck prisms were used as a safe way to illuminate decks on ships. The deck prism shown here is composed of the following three solids: a regular hexagonal prism with an edge length of 3.5 inches and a height of 1.5 inches, a regular hexagonal prism with an edge length of 3.25 inches and a height of 0.25 inch, and a regular hexagonal pyramid with an edge length of 3 inches and a height of 3 inches. Find the volume of the deck prism. A C 3 ft 41. CRITICAL THINKING When the given triangle is rotated around each of its sides, solids of revolution are formed. Describe the three solids and find their volumes. Give your answers in terms of π. 38. THOUGHT PROVOKING A frustum of a cone is the part of the cone that lies between the base and a plane parallel to the base, as shown. Write a formula for the volume of the frustum of a cone in terms of a, b, and h. (Hint: Consider the missing top of the cone and use similar triangles.) h a b CRITICAL THINKING A square pyramid is inscribed in a right cylinder so that the base of the pyramid is on a base of the cylinder, and the vertex of the pyramid is on the other base of the cylinder. The cylinder has a radius of 6 feet and a height of 12 feet. Find the volume of the pyramid Maintaining Mathematical Proficiency Find the indicated measure. (Section 11.2) Reviewing what you learned in previous grades and lessons 43. area of a circle with a radius of 7 feet 44. area of a circle with a diameter of 22 centimeters 45. diameter of a circle with an area of 256π 46. radius of a circle with an area of 529π 678 Chapter 12 Surface Area and Volume

44 12.6 Surface Areas and Volumes of Spheres TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10. G.11.C G.11.D Essential Question How can you find the surface area and the volume of a sphere? Finding the Surface Area of a Sphere Work with a partner. Remove the covering from a baseball or softball. r SELECTING TOOLS To be proficient in math, you need to identify relevant external mathematical resources, such as content located on a website. You will end up with two figure 8 pieces of material, as shown above. From the amount of material it takes to cover the ball, what would you estimate the surface area S of the ball to be? Express your answer in terms of the radius r of the ball. S= Surface area of a sphere Use the Internet or some other resource to confirm that the formula you wrote for the surface area of a sphere is correct. Finding the Volume of a Sphere Work with a partner. A cylinder is circumscribed about a sphere, as shown. Write a formula for the volume V of the cylinder in terms of the radius r. V= r r Volume of cylinder 2r When half of the sphere (a hemisphere) is filled with sand and poured into the cylinder, it takes three hemispheres to fill the cylinder. Use this information to write a formula for the volume V of a sphere in terms of the radius r. V= Volume of a sphere Communicate Your Answer 3. How can you find the surface area and the volume of a sphere? 4. Use the results of Explorations 1 and 2 to find the surface area and the volume of a sphere with a radius of (a) 3 inches and (b) 2 centimeters. Section 12.6 Surface Areas and Volumes of Spheres 679

45 12.6 Lesson What You Will Learn Core Vocabulary chord of a sphere, p. 680 great circle, p. 680 Previous sphere center of a sphere radius of a sphere diameter of a sphere hemisphere Find surface areas of spheres. Find volumes of spheres. Finding Surface Areas of Spheres A sphere is the set of all points in space equidistant from a given point. This point is called the center of the sphere. A radius of a sphere is a segment from the center to a point on the sphere. A chord of a sphere is a segment whose endpoints are on the sphere. A diameter of a sphere is a chord that contains the center. center C radius chord C diameter As with circles, the terms radius and diameter also represent distances, and the diameter is twice the radius. If a plane intersects a sphere, then the intersection is either a single point or a circle. If the plane contains the center of the sphere, then the intersection is a great circle of the sphere. The circumference of a great circle is the circumference of the sphere. Every great circle of a sphere separates the sphere into two congruent halves called hemispheres. great circle hemispheres Core Concept Surface Area of a Sphere The surface area S of a sphere is S = 4πr 2 where r is the radius of the sphere. r S = 4 πr 2 To understand the formula for the surface area of a sphere, think of a baseball. The surface area of a baseball is sewn from two congruent shapes, each of which resembles two joined circles. So, the entire covering of the baseball consists of four circles, each with radius r. The area A of a circle with radius r is A = πr 2. So, the area of the covering can be approximated by 4πr 2. This is the formula for the surface area of a sphere. leather covering r 680 Chapter 12 Surface Area and Volume

46 Find the surface area of each sphere. Finding the Surface Areas of Spheres a. 8 in. b. C = 12 π ft a. S= 4πr 2 Formula for surface area of a sphere = 4π(8) 2 Substitute 8 for r. = 256π Simplify Use a calculator. The surface area is 256π, or about square inches. b. The circumference of the sphere is 12π, so the radius of the sphere is 12π 2π = 6 feet. S = 4πr 2 Formula for surface area of a sphere = 4π(6) 2 Substitute 6 for r. = 144π Simplify Use a calculator. The surface area is 144π, or about square feet. Find the diameter of the sphere. Finding the Diameter of a Sphere S = 4πr 2 Formula for surface area of a sphere COMMON ERROR e sure to multiply the value of r by 2 to find the diameter π = 4πr 2 Substitute 20.25π for S = r 2 Divide each side by 4π = r Find the positive square root. The diameter is 2r = = 4.5 centimeters. S = π cm 2 Monitoring Progress Help in English and Spanish at igideasmath.com Find the surface area of the sphere ft 2. C = 6 π ft 3. Find the radius of the sphere. S = 30 π m 2 Section 12.6 Surface Areas and Volumes of Spheres 681

47 Finding Volumes of Spheres The figure shows a hemisphere and a cylinder with a cone removed. A plane parallel to their bases intersects the solids z units above their bases. r 2 z 2 z r r r Using the AA Similarity Theorem (Theorem 8.3), you can show that the radius of the cross section of the cone at height z is z. The area of the cross section formed by the plane is π(r 2 z 2 ) for both solids. ecause the solids have the same height and the same cross-sectional area at every level, they have the same volume by Cavalieri s Principle. V hemisphere = V cylinder V cone = πr 2 (r) 1 3 πr2 (r) = 2 3 πr3 So, the volume of a sphere of radius r is 2 V hemisphere = πr3 = 4 3 πr3. Core Concept Volume of a Sphere The volume V of a sphere is V = 4 3 πr3 where r is the radius of the sphere. r 4 V = πr 3 3 Finding the Volume of a Sphere Find the volume of the soccer ball. 4.5 in. V = 4 3 πr3 Formula for volume of a sphere = 4 3 π(4.5)3 Substitute 4.5 for r. = 121.5π Simplify Use a calculator. The volume of the soccer ball is 121.5π, or about cubic inches. 682 Chapter 12 Surface Area and Volume

48 Changing Dimensions in a Solid Describe how multiplying the radius by 1 4 affects the volume of the sphere. 12 in. efore change After change Dimensions r = 12 in. r = 3 in. Volume V = 4 3 πr3 = 4 3 π(12)3 = 2304π in. 3 V = 4 3 πr3 = 4 3 π(3)3 = 36π in. 3 Multiplying the radius by 1 4 results in a volume that is 36π 2304π = 1 64 ( = 1 times the original volume. 4) 3 Find the volume of the composite solid. Finding the Volume of a Composite Solid 2 in. Volume of solid = Volume of cylinder Volume of hemisphere 2 in. = πr 2 h 1 2 ( 4 3 πr3 ) Write formulas. = π(2) 2 (2) 2 3 π(2)3 Substitute. = 8π 16 3 π Multiply. = 24 3 π 16 3 π Rewrite fractions using least common denominator. = 8 3 π Subtract The volume is 8 π, or about 8.38 cubic inches. 3 Use a calculator. 1 m 5 m Monitoring Progress Help in English and Spanish at igideasmath.com 4. The radius of a sphere is 5 yards. Find the volume of the sphere. 5. The diameter of a sphere is 36 inches. Find the volume of the sphere. 6. In Example 4, describe how doubling the radius affects the volume of the sphere. 7. A sphere has a radius of 18 centimeters. Describe how multiplying the radius by 1 affects the volume of the sphere Find the volume of the composite solid at the left. Section 12.6 Surface Areas and Volumes of Spheres 683

49 12.6 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. VOCAULARY When a plane intersects a sphere, what must be true for the intersection to be a great circle? 2. WRITING Explain the difference between a sphere and a hemisphere. Monitoring Progress and Modeling with Mathematics In Exercises 3 6, find the surface area of the sphere. (See Example 1.) In Exercises 13 18, find the volume of the sphere. (See Example 3.) 3. 4 ft cm m ft m C = 4 π ft 22 yd 14 ft In Exercises 7 10, find the indicated measure. (See Example 2.) 7. Find the radius of a sphere with a surface area of 4π square feet C = 20 cm π C = 7 π in. 8. Find the radius of a sphere with a surface area of 1024π square inches. 9. Find the diameter of a sphere with a surface area of 900π square meters. 10. Find the diameter of a sphere with a surface area of 196π square centimeters. In Exercises 11 and 12, find the surface area of the hemisphere m in. In Exercises 19 and 20, find the volume of the sphere with the given surface area. 19. Surface area = 16π ft Surface area = 484π cm ERROR ANALYSIS Describe and correct the error in finding the volume of the sphere. 6 ft V = 4 3 π(6)2 = 48π ft Chapter 12 Surface Area and Volume

50 22. ERROR ANALYSIS Describe and correct the error in finding the volume of the sphere. 3 in. V = 4 3 π(3)3 = 36π in. 3 In Exercises 23 and 24, describe how the change affects the volume of the sphere. (See Example 4.) 33. MAKING AN ARGUMENT Your friend claims that if the radius of a sphere is doubled, then the surface area of the sphere will also be doubled. Is your friend correct? Explain your reasoning. 34. REASONING A semicircle with a diameter of 18 inches is rotated about its diameter. Find the surface area and the volume of the solid formed. 35. MODELING WITH MATHEMATICS A silo has the dimensions shown. The top of the silo is a hemispherical shape. Find the volume of the silo. 23. tripling the radius 24. multiplying the radius by m 36 cm 60 ft 20 ft In Exercises 25 28, find the volume of the composite solid. (See Example 5.) in. 5 in ft 12 ft 36. MODELING WITH MATHEMATICS Three tennis balls are stored in a cylindrical container with a height of 8 inches and a radius of 1.43 inches. The circumference of a tennis ball is 8 inches. a. Find the volume of a tennis ball cm m 10 cm 6 m b. Find the amount of space within the cylinder not taken up by the tennis balls. 37. ANALYZING RELATIONSHIPS Use the table shown for a sphere. Radius Surface area Volume In Exercises 29 32, find the surface area and volume of the ball. 29. bowling ball 30. basketball 3 in. 36π in. 2 36π in. 3 6 in. 9 in. 12 in. a. Copy and complete the table. Leave your answers in terms of π. d = 8.5 in. C = 29.5 in. 31. softball 32. golf ball b. What happens to the surface area of the sphere when the radius is doubled? tripled? quadrupled? c. What happens to the volume of the sphere when the radius is doubled? tripled? quadrupled? C = 12 in. d = 1.7 in. 38. MATHEMATICAL CONNECTIONS A sphere has a diameter of 4(x + 3) centimeters and a surface area of 784π square centimeters. Find the value of x. Section 12.6 Surface Areas and Volumes of Spheres 685

51 39. MODELING WITH MATHEMATICS The radius of Earth is about 3960 miles. The radius of the moon is about 1080 miles. a. Find the surface area of Earth and the moon. b. Compare the surface areas of Earth and the moon. c. About 70% of the surface of Earth is water. How many square miles of water are on Earth s surface? 40. MODELING WITH MATHEMATICS The Torrid Zone on Earth is the area between the Tropic of Cancer and the Tropic of Capricorn. The distance between these two tropics is about 3250 miles. You can estimate the distance as the height of a cylindrical belt around the Earth at the equator. Tropic of Cancer 43. CRITICAL THINKING Let V be the volume of a sphere, S be the surface area of the sphere, and r be the radius of the sphere. Write an equation for V in terms of r and S. ( Hint: Start with the ratio V S ). 44. THOUGHT PROVOKING A spherical lune is the region between two great circles of a sphere. Find the formula for the area of a lune. r θ 3250 mi equator Torrid Zone Tropic of Capricorn 45. CRITICAL THINKING The volume of a right cylinder is the same as the volume of a sphere. The radius of the sphere is 1 inch. Give three possibilities for the dimensions of the cylinder. a. Estimate the surface area of the Torrid Zone. (The radius of Earth is about 3960 miles.) b. A meteorite is equally likely to hit anywhere on Earth. Estimate the probability that a meteorite will land in the Torrid Zone. 41. ASTRACT REASONING A sphere is inscribed in a cube with a volume of 64 cubic inches. What is the surface area of the sphere? Explain your reasoning. 42. HOW DO YOU SEE IT? The formula for the volume of a hemisphere and a cone are shown. If each solid has the same radius and r = h, which solid will have a greater volume? Explain your reasoning. r r 46. PROLEM SOLVING A spherical cap is a portion of a sphere cut off by a plane. The formula for the volume of a spherical cap is V = πh 6 (3a2 + h 2 ), where a is the radius of the base of the cap and h is the height of the cap. Use the diagram and given information to find the volume of each spherical cap. h a. r = 5 ft, a = 4 ft b. r = 34 cm, a = 30 cm c. r = 13 m, h = 8 m d. r = 75 in., h = 54 in. a r 2 V = πr 3 3 h 1 V = πr 2 h 3 Maintaining Mathematical Proficiency Use the diagram. (Section 1.1) 48. Name four points. 49. Name two line segments. 50. Name two lines. 51. Name a plane. 47. CRITICAL THINKING A sphere with a radius of 2 inches is inscribed in a right cone with a height of 6 inches. Find the surface area and the volume of the cone. Reviewing what you learned in previous grades and lessons R U S T P Q 686 Chapter 12 Surface Area and Volume

52 12.7 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.4.D Spherical Geometry Essential Question How can you represent a line on a sphere? The endpoints of a diameter of a sphere are called antipodal points. A great circle Points A and are antipodal. ANALYZING MATHEMATICAL RELATIONSHIPS To be proficient in math you need to look closely to discern a pattern or structure. Finding the Shortest Distance etween Two Points Work with a partner. Use a washable marker, a piece of paper, a ball, and a piece of string. a. Draw two points on the piece of paper. Label the points A and. Use the string to create the shortest path between the points. What geometric term describes the path of the string? When you extend the path infinitely in either direction, what geometric term describes the path? b. Draw two points that are not antipodal on the ball. Label the points C and D. Use the string to create the shortest C path between the points. What geometric term describes D the path of the string? Can you extend the path infinitely in either direction? What geometric term describes the path when you extend it all the way around the ball? c. Compare and contrast the paths on the piece of paper in part (a) and the paths on the ball in part (b). Exploring Lines on a Sphere Work with a partner. Use a ball and three rubber bands. a. Construct a great circle on the ball using a rubber band. b. Construct a second distinct great circle. At how many points do the two great circles intersect? Compare this to the number of points at which two distinct lines can intersect in the plane. c. How many great circles can you construct through two points that are not antipodal? Explain. d. How many great circles can you construct through two points that are antipodal? Explain. e. Is the Two Point Postulate (Postulate 2.1) true for great circles on a sphere? Explain. Two Point Postulate: Through any two points, there exists exactly one line. f. A polygon on the surface of a sphere is a shape enclosed by arcs of great circles. Describe how you can construct a polygon with two sides on the surface of a sphere. Then construct a two-sided polygon on the ball using rubber bands. g. Describe how you can construct a triangle on the surface of a sphere. Then construct a triangle on the ball using rubber bands. Communicate Your Answer 3. How can you represent a line on a sphere? Section 12.7 Spherical Geometry 687

53 12.7 Lesson What You Will Learn Core Vocabulary antipodal points, p. 688 Previous great circle Compare Euclidean and spherical geometry. Find distances on a sphere. Find areas of spherical triangles. Comparing Euclidean and Spherical Geometry In Euclidean geometry, a plane is a flat surface that extends without end in all directions, and a line in the plane is a set of points that extends without end in two directions. Geometry on a sphere is different. In spherical geometry, a plane is the surface of a sphere, a line is a great circle, and the angle between two lines is the angle between the planes containing the two corresponding great circles. Core Concept Euclidean Geometry and Spherical Geometry Euclidean Geometry center great circles Spherical Geometry A center P A S m Plane P contains line and point A not on the line. Sphere S contains great circle m and point A not on m. Great circle m is a line. A A C P S C The vertices of AC are points in plane P and the sides are segments. The sum of the interior angles of a triangle is 180. m A + m + m C = 180 The vertices of AC are points on sphere S and the sides are arcs of great circles. The sum of the interior angles of a spherical triangle is greater than 180. m A + m + m C > 180 Some properties and postulates in Euclidean geometry are true in spherical geometry. Others are not, or are true only under certain circumstances. For example, in Euclidean geometry, the Two Point Postulate (Postulate 2.1) states that through any two points, there exists exactly one line. In spherical geometry, this postulate is true only for points that are not the endpoints of a diameter of the sphere. The endpoints of a diameter of a sphere are called antipodal points. 688 Chapter 12 Surface Area and Volume

54 Comparing Euclidean and Spherical Geometry Tell whether the following postulate in Euclidean geometry is also true in spherical geometry. Draw a diagram to support your answer. Parallel Postulate (Postulate 3.1): If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. Parallel lines do not intersect. The sphere shows a line (a great circle) and a point A not on. Several lines are drawn through A. Each great circle containing A intersects. So, there can be no line parallel to. The Parallel Postulate is not true in spherical geometry. A Monitoring Progress Help in English and Spanish at igideasmath.com 1. Draw sketches to show that the Two Point Postulate (Postulate 2.1), discussed on the previous page, is not true for antipodal points and is true for two points that are not antipodal points. Finding Distances on a Sphere In Euclidean geometry, there is exactly one distance that can be measured between any two points. On a sphere, there are two distances that can be measured between any two points. These distances are the lengths of the major and minor arcs of the great circle drawn through the points. Finding Distances on a Sphere The diameter of the sphere is 15 inches, and m A = 60. Find the distances between 15 in. A C points A and. P Find the lengths of the minor arc A and the major arc AC of the great circle shown. Let x be the arc length of A and let y be the arc length of AC. Arc length of A = m A Arc length of AC = m AC 2πr 360 2πr 360 x 15π = 60 y = π 360 x = 2.5π y = 12.5π x 7.85 y The distances are about 7.85 inches and about inches. A 20 cm P C Monitoring Progress Help in English and Spanish at igideasmath.com 2. The diameter of the sphere is 20 centimeters, and m A = 120. Find the distances between points A and. Section 12.7 Spherical Geometry 689

55 Finding Areas of Spherical Triangles Core Concept Area of a Spherical Triangle The area of AC on a sphere is A = πr2 (m A + m + m C 180 ) 180 where r is the radius of the sphere. A r C Finding Areas of Spherical Triangles Find the area of each spherical triangle. a. DEF D 95 b. JKL 10 in. J m E F 90 K 78 L a. A = πr2 (m D + m E + m F 180 ) Formula for area of a spherical triangle 180 = π (10)2 ( ) 180 Substitute. = 100π (135 ) 180 Simplify. = 75π Simplify Use a calculator. The area of DEF is 75π, or about square inches. b. A = πr2 (m J + m K + m L 180 ) Formula for area of a spherical triangle 180 = π(12)2 ( ) 180 Substitute. = 144π (90 ) 180 Simplify. = 72π Simplify Use a calculator. The area of JKL is 72π, or about square meters. Monitoring Progress Find the area of the spherical triangle. Help in English and Spanish at igideasmath.com 3. MNP 4. QRS 5. TUV M N P 8 cm R Q mm S T U V 12 m 690 Chapter 12 Surface Area and Volume

56 12.7 Exercises Tutorial Help in English and Spanish at igideasmath.com Vocabulary and Core Concept Check 1. WRITING How is a line in Euclidean geometry different from a line in spherical geometry? 2. WHICH ONE DOESN T ELONG? Which statement does not belong with the other three? Explain your reasoning. There are no parallel lines. Two distances can be measured between any two points. The sum of the interior angles of a triangle is greater than 180. A line extends without end. Monitoring Progress and Modeling with Mathematics In Exercises 3 8, the statement is true in Euclidean geometry. Rewrite the statement to be true for spherical geometry. Explain your reasoning. (See Example 1.) 3. Given a line and a point not on the line, there is exactly one line through the point parallel to the given line. 4. If two lines intersect, then their intersection is exactly one point. 5. The length of a line is infinite. 13. m A = 150 A 14 yd P C 14. m A = 135 A 40 in. P In Exercises 15 20, find the area of the spherical triangle. (See Example 3.) C 6. A line divides a plane into two infinite regions. 7. A triangle can have no more than one right angle. 8. Two distinct lines in a plane are either parallel or intersect at exactly one point. 15. AC 16. DEF m A C 90 E D F 6 ft In Exercises 9 14, use the diagram and the given arc measure to find the distances between points A and. (See Example 2.) 9. m A = 90 A 16 cm P 11. m A = 30 A 12 ft P C C 10. m A = 140 A 30 mm P 12. m A = 45 A 18 m P C C 17. JKL 18. MNP K 75 M J in L 120 N 19. QRS 20. TUV R 140 Q 16 mm S U P T cm 119 V 5 yd Section 12.7 Spherical Geometry 691

57 21. ERROR ANALYSIS Describe and correct the error in finding the distances between points A and. Arc length of A = m A 2πr 360 x 2π(24) = x 24 cm 48π = 90 A C 360 P x = 12π Arc length of AC = m AC 2πr 360 y = 2π(24) 360 y 48π = The distances are about centimeters and about centimeters. y = 36π REASONING Determine whether each of the theorems from Euclidean geometry is true in spherical geometry. Explain your reasoning. a. Linear Pair Perpendicular Theorem (Theorem 3.10): If two lines intersect to form a linear pair of congruent angles, then the lines are perpendicular. b. Lines Perpendicular to a Transversal Theorem (Theorem 3.12): In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. 25. MODELING WITH MATHEMATICS The radius of Earth is about 3960 miles. Find the distance between Pontianak, Indonesia, and the North Pole. Pontianak, Indonesia North Pole Equator 22. ERROR ANALYSIS Describe and correct the error in finding the area of AC. πr A = 2 ( m A + m + m C ) 180 = π(5)2 ( ) π = ( 180 )324 A = 45π in The area of AC C is 45π, or about square inches. 23. MAKING AN ARGUMENT A polygon on a sphere has arcs of great circles for sides. Your friend claims that a polygon on a sphere can have two sides. Is your friend correct? Explain your reasoning. 26. HOW DO YOU SEE IT? In Euclidean geometry, when three points are collinear, exactly one of the points lies between the other two. Is this true in spherical geometry? Explain. A C 27. REASONING In Euclidean geometry, two perpendicular lines form four right angles. How many right angles do two perpendicular lines form in spherical geometry? 28. THOUGHT PROVOKING What are the possible sums of the interior angles of a spherical triangle? Explain. A C m Maintaining Mathematical Proficiency Find the areas of the sectors formed by DFE. (Section 11.2) 29. G D F 5 ft 120 E 30. F 8 yd 45 D E G Reviewing what you learned in previous grades and lessons 31. D G 6 m E F Chapter 12 Surface Area and Volume

58 What Did You Learn? Core Vocabulary volume, p. 664 Cavalieri s Principle, p. 664 chord of a sphere, p. 680 great circle, p. 680 antipodal points, p. 688 Core Concepts Section 12.4 Cavalieri s Principle, p. 664 Volume of a Prism, p. 664 Volume of a Cylinder, p. 665 Section 12.5 Volume of a Pyramid, p. 672 Volume of a Cone, p. 673 Section 12.6 Surface Area of a Sphere, p. 680 Volume of a Sphere, p. 682 Section 12.7 Euclidean Geometry and Spherical Geometry, p. 688 Finding Distances on a Sphere, p. 689 Area of a Spherical Triangle, p. 690 Mathematical Thinking 1. Search online for advertisements for products that come in different sizes. Then compare the unit prices, as done in Exercise 36 on page 670. Do you get results similar to Exercise 36? Explain. 2. In Exercise 30 on page 677, the cube is formed by three oblique pyramids. If these pyramids were right pyramids, it would not be possible to form a cube. Explain why the formula for the volume of a right pyramid is the same as the formula for the volume of an oblique pyramid. 3. In Exercise 38 on page 685, explain the steps you used to find the value of x. Water Park Renovation Performance e Task The city council will consider reopening the closed water park if your team can come up with a cost analysis for painting some of the structures, filling the pool water reservoirs, and resurfacing some of the surfaces. What is your plan to convince the city council to open the water park? To explore the answer to this question and more, go to igideasmath.com. 693

59 12 Chapter Review 12.1 Three-Dimensional Figures (pp ) Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid The solid is a cylinder with a height of 8 and a radius of Sketch the solid produced by rotating the figure around the given axis. Then identify and describe the solid Describe the cross section formed by the intersection of the plane and the solid Surface Areas of Prisms and Cylinders (pp ) Find the lateral area and the surface area of the right rectangular prism. L = Ph Formula for lateral area of a right prism = [2(8) + 2(14)](9) Substitute. = 396 Simplify. S = 2 + L Formula for surface area of a right prism = 2(8)(14) Substitute. 9 in. 8 in. 14 in. = 620 Simplify. The lateral area is 396 square inches and the surface area is 620 square inches. 694 Chapter 12 Surface Area and Volume

60 Find the lateral area and the surface area of the right prism or right cylinder m 18 cm 4 cm 17 ft 16 ft 20 ft 11 m 12.3 Surface Areas of Pyramids and Cones (pp ) Find the lateral area and the surface area of the right cone. 15 m Use the Pythagorean Theorem (Theorem 9.1) to find the slant height. = = 39 Find the lateral area and the surface area. L = πr Formula for lateral area of a right cone = π(15)(39) Substitute. = 585π Simplify Use a calculator. 36 m S = πr 2 + πr = π(15) π Substitute. = 810π Simplify Formula for surface area of a right cone Use a calculator. The lateral area is 585π, or about square meters. The surface area is 810π, or about square meters. Find the lateral area and the surface area of the regular pyramid or right cone in cm ft 9 in. 10 cm 3 ft 6 ft 13. Find the lateral area and the surface area of the composite solid. 4 cm 3 cm 12 cm Chapter 12 Chapter Review 695

61 12.4 Volumes of Prisms and Cylinders (pp ) Find the volume of the cylinder. The dimensions of the cylinder are r = 7 ft and h = 10 ft. 7 ft V = πr 2 h = π(7) 2 (10) Substitute. Formula for volume of a cylinder 10 ft = 490π Simplify Use a calculator. The volume is 490π, or about cubic feet. Find the volume of the solid m 8 mm 1.5 m 2.1 m 2 mm 2 yd 4 yd 17. Describe how the change affects the volume of the triangular prism. a. multiplying the height of the prism by 1 3 b. multiplying all the linear dimensions by 2 6 in. 6 in. 10 in. 7 cm 24 cm 30 cm 6 in. 18. Find the volume of the composite solid Volumes of Pyramids and Cones (pp ) Find the volume of the pyramid. V = 1 h Formula for volume of a pyramid 3 = 1 3 ( ) (12) Substitute. 12 m = 80 Simplify. The volume is 80 cubic meters. 5 m 8 m 696 Chapter 12 Surface Area and Volume

62 Find the volume of the solid cm 30 cm 16 cm m 18 m m 13 m 10 m 22. The volume of a square pyramid is 1024 cubic inches. The base has a side length of 16 inches. Find the height of the pyramid. 23. A cone with a diameter of 16 centimeters has a volume of 320π cubic centimeters. Find the height of the cone Surface Areas and Volumes of Spheres (pp ) Find the (a) surface area and (b) volume of the sphere. a. S = 4πr 2 Formula for surface area of a sphere = 4π(18) 2 Substitute 18 for r. = 1296π Simplify Use a calculator. 18 in. The surface area is 1296π, or about square inches. b. V = 4 3 πr3 Formula for volume of a sphere = 4 3 π(18)3 Substitute 18 for r. = 7776π Simplify. 24, Use a calculator. The volume is 7776π, or about 24, cubic inches. Find the surface area and the volume of the sphere in. 17 ft C = 30 π ft 27. The shape of Mercury can be approximated by a sphere with a diameter of 4880 kilometers. Find the surface area and the volume of Mercury. 28. A solid is composed of a cube with a side length of 6 meters and a hemisphere with a diameter of 6 meters. Find the volume of the composite solid. Chapter 12 Chapter Review 697

63 12.7 Spherical Geometry (pp ) a. The diameter of the sphere is 24 inches, and m A = 30. Find the distances between points A and. Find the lengths of the minor arc A and the major arc AC of the great circle shown. Let x be the arc length of A and let y be the arc length of AC. Arc length of A = m A Arc length of AC = m AC 2πr 360 x 24π = x = 2π 2πr y 24π = 360 y = 22π x 6.28 y The distances are about 6.28 inches and about inches. A 24 in. P C b. Find the area of DEF. A = πr2 180 (m D + m E + m F 180 ) Formula for area of a spherical triangle = π (6)2 ( ) Substitute. 180 = 36π (100 ) Simplify. 180 = 20π Simplify. E D ft F Use a calculator. The area of DEF is 20π, or about square feet. X 18 ft P Y Z 29. The diameter of the sphere is 18 feet, and m XY = 90. Find the distances between points X and Y. Find the area of the spherical triangle. 30. JKL 31. RST 32. XYZ K J in L S R m T X cm Y Z 698 Chapter 12 Surface Area and Volume

64 12 Chapter Test Find the volume of the solid m 4 m 4. 4 ft 15.5 m 3.2 ft 6 m 8 ft 8 m 3 m 4 m 5 ft 2 ft Find the lateral area and the surface area of the solid m 8.3 m 6. 9 ft 7. 4 in cm 16 cm 12 m 20 ft 6 in. 7 in. 30 cm 7 in. 9. Sketch the composite solid produced by rotating the figure around the given axis. Then identify and describe the composite solid. 10. The shape of Mars can be approximated by a sphere with a diameter of 6779 kilometers. Find the surface area and the volume of Mars You have a funnel with the dimensions shown. a. Find the approximate volume of the funnel. b. You use the funnel to put oil in a car. Oil flows out of the funnel at a rate of 45 milliliters per second. How long will it take to empty the funnel when it is full of oil? (1 ml = 1 cm 3 ) c. How long would it take to empty a funnel with a radius of 10 centimeters and a height of 6 centimeters if oil flows out of the funnel at a rate of 45 milliliters per second? d. Explain why you can claim that the time calculated in part (c) is greater than the time calculated in part (b) without doing any calculations. 6 cm 10 cm 12. Describe how the change affects the surface area and the volume of the right cone. a. multiplying the height by 3 4 b. multiplying all the linear dimensions by yd 13. A water bottle in the shape of a cylinder has a volume of 500 cubic centimeters. The diameter of a base is 7.5 centimeters. What is the height of the bottle? Justify your answer. 15 yd 14. Is it possible for a right triangular pyramid to have a cross section that is a quadrilateral? If so, describe or sketch how the plane could intersect the pyramid. 15. The soccer ball shown has a radius of 4.5 inches. Each interior angle of the spherical triangles has a measure of 70. Find the area of one spherical triangle. Chapter 12 Chapter Test 699

65 12 Standards Assessment 1. Which cross section formed by the intersection of the plane and the solid is a rectangle? (TEKS G.10.A) A C D 2. The coordinates of the vertices of DEF are D( 8, 5), E( 5, 8), and F( 1, 4). Which set of coordinates describes a triangle that is similar to DEF? (TEKS G.7.A) F A( 24, 15), ( 15, 24), and C( 3, 8) G J(16, 10), K(10, 16), and L(2, 8) H P( 8, 10), Q( 5, 16), and R( 1, 8) J U( 10, 5), V( 7, 2), and W( 2, 7) 3. The top of the Washington Monument in Washington, D.C., is a square pyramid, called a pyramidion. What is the volume of the pyramidion? (TEKS G.11.D) A 22, ft 3 172, ft 3 C 66, ft 3 D 207, ft ft 34.5 ft 4. Use the diagram. Which proportion is false? (TEKS G.8.) F D DC = DA D G CA A = A AD D C H CA A = A CA J DC C = C CA A 700 Chapter 12 Surface Area and Volume

66 5. The surface area of the right cone is 200π square feet. What is the slant height of the cone? (TEKS G.11.C) A 10.5 ft 17 ft C 23 ft D 24 ft 16 ft 6. In the diagram, ACD is a parallelogram. Which statement is not true? (TEKS G.6.) A E D C F AED CE G DEC EA H AD CD J AE DEC 7. Points X, Y, and Z lie on the surface of a sphere. What is a possible value of the sum of the interior angles of XYZ? (TEKS G.4.D) A C 180 D GRIDDED ANSWER The diagram shows a square pyramid and a cone. oth solids have a height of 23 inches, and the base of the cone has a radius of 8 inches. According to Cavalieri s Principle, the solids will have the same volume if the square base has sides of length inches. Round your answer to the nearest tenth of an inch. (TEKS G.11.D) h r 9. Which statement about the circle is not true? (TEKS G.12.A) F AD is a diameter. G H J H is a chord. AD is a chord. CD is a radius. A G H C F D E Chapter 12 Standards Assessment 701