12 Surface Area and Volume


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1 12 Surface Area and Volume 12.1 ThreeDimensional 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 ThreeDimensional Figures A net for a threedimensional figure is a twodimensional pattern that can be folded to form the threedimensional 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 threedimensional 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 ThreeDimensional 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 ThreeDimensional 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 threedimensional 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 ThreeDimensional Figures 641
7 Sketching and Describing Solids of Revolution A solid of revolution is a threedimensional figure that is formed by rotating a twodimensional 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 ThreeDimensional 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 twodimensional 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 RealLife 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 cubeshaped 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 RealLife 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 3inch 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 crosssectional area at every level, then they have the same volume. The prisms below have equal heights h and equal crosssectional 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 crosssectional 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
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