12 Surface Area and Volume

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "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 = + 12Pā„“ = (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 = š›‘r 2 + š›‘r 2ā„“ = š›‘ (6)2 + š›‘ (6)2(10) 6 cm = 396š›‘ 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

Surface Areas of Prisms and Cylinders

Surface Areas of Prisms and Cylinders 12.2 TEXAS ESSENTIAL KNOWLEDGE AND SKILLS G.10.B G.11.C Surface Areas of Prisms and Cylinders Essential Question How can you find te surface area of a prism or a cylinder? Recall tat te surface area of

More information

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources:

Geometry Honors: Extending 2 Dimensions into 3 Dimensions. Unit Overview. Student Focus. Semester 2, Unit 5: Activity 30. Resources: Online Resources: Geometry Honors: Extending 2 Dimensions into 3 Dimensions Semester 2, Unit 5: Activity 30 Resources: SpringBoard- Geometry Online Resources: Geometry Springboard Text Unit Overview In this unit students

More information

Junior Math Circles March 10, D Geometry II

Junior Math Circles March 10, D Geometry II 1 University of Waterloo Faculty of Mathematics Junior Math Circles March 10, 2010 3D Geometry II Centre for Education in Mathematics and Computing Opening Problem Three tennis ball are packed in a cylinder.

More information

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid

SA B 1 p where is the slant height of the pyramid. V 1 3 Bh. 3D Solids Pyramids and Cones. Surface Area and Volume of a Pyramid Accelerated AAG 3D Solids Pyramids and Cones Name & Date Surface Area and Volume of a Pyramid The surface area of a regular pyramid is given by the formula SA B 1 p where is the slant height of the pyramid.

More information

Area of Parallelograms, Triangles, and Trapezoids (pages 314 318)

Area of Parallelograms, Triangles, and Trapezoids (pages 314 318) Area of Parallelograms, Triangles, and Trapezoids (pages 34 38) Any side of a parallelogram or triangle can be used as a base. The altitude of a parallelogram is a line segment perpendicular to the base

More information

10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warm-up

10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres. 10.4 Day 1 Warm-up 10.4 Surface Area of Prisms, Cylinders, Pyramids, Cones, and Spheres 10.4 Day 1 Warm-up 1. Which identifies the figure? A rectangular pyramid B rectangular prism C cube D square pyramid 3. A polyhedron

More information

CK-12 Geometry: Exploring Solids

CK-12 Geometry: Exploring Solids CK-12 Geometry: Exploring Solids Learning Objectives Identify different types of solids and their parts. Use Euler s formula to solve problems. Draw and identify different views of solids. Draw and identify

More information

SURFACE AREA AND VOLUME

SURFACE AREA AND VOLUME SURFACE AREA AND VOLUME In this unit, we will learn to find the surface area and volume of the following threedimensional solids:. Prisms. Pyramids 3. Cylinders 4. Cones It is assumed that the reader has

More information

Area of Parallelograms (pages 546 549)

Area of Parallelograms (pages 546 549) A Area of Parallelograms (pages 546 549) A parallelogram is a quadrilateral with two pairs of parallel sides. The base is any one of the sides and the height is the shortest distance (the length of a perpendicular

More information

12 Surface Area and Volume

12 Surface Area and Volume CHAPTER 12 Surface Area and Volume Chapter Outline 12.1 EXPLORING SOLIDS 12.2 SURFACE AREA OF PRISMS AND CYLINDERS 12.3 SURFACE AREA OF PYRAMIDS AND CONES 12.4 VOLUME OF PRISMS AND CYLINDERS 12.5 VOLUME

More information

Integrated Algebra: Geometry

Integrated Algebra: Geometry Integrated Algebra: Geometry Topics of Study: o Perimeter and Circumference o Area Shaded Area Composite Area o Volume o Surface Area o Relative Error Links to Useful Websites & Videos: o Perimeter and

More information

Geometry Unit 6 Areas and Perimeters

Geometry Unit 6 Areas and Perimeters Geometry Unit 6 Areas and Perimeters Name Lesson 8.1: Areas of Rectangle (and Square) and Parallelograms How do we measure areas? Area is measured in square units. The type of the square unit you choose

More information

Perfume Packaging. Ch 5 1. Chapter 5: Solids and Nets. Chapter 5: Solids and Nets 279. The Charles A. Dana Center. Geometry Assessments Through

Perfume Packaging. Ch 5 1. Chapter 5: Solids and Nets. Chapter 5: Solids and Nets 279. The Charles A. Dana Center. Geometry Assessments Through Perfume Packaging Gina would like to package her newest fragrance, Persuasive, in an eyecatching yet cost-efficient box. The Persuasive perfume bottle is in the shape of a regular hexagonal prism 10 centimeters

More information

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry

Surface Area of Rectangular & Right Prisms Surface Area of Pyramids. Geometry Surface Area of Rectangular & Right Prisms Surface Area of Pyramids Geometry Finding the surface area of a prism A prism is a rectangular solid with two congruent faces, called bases, that lie in parallel

More information

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier

SOLID SHAPES M.K. HOME TUITION. Mathematics Revision Guides Level: GCSE Higher Tier Mathematics Revision Guides Solid Shapes Page 1 of 19 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier SOLID SHAPES Version: 2.1 Date: 10-11-2015 Mathematics Revision Guides Solid

More information

Name: Date: Geometry Honors Solid Geometry. Name: Teacher: Pd:

Name: Date: Geometry Honors Solid Geometry. Name: Teacher: Pd: Name: Date: Geometry Honors 2013-2014 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 1-6 HW: Pgs: 7-10 DAY 2: SWBAT: Calculate the Volume

More information

Shape Dictionary YR to Y6

Shape Dictionary YR to Y6 Shape Dictionary YR to Y6 Guidance Notes The terms in this dictionary are taken from the booklet Mathematical Vocabulary produced by the National Numeracy Strategy. Children need to understand and use

More information

Height. Right Prism. Dates, assignments, and quizzes subject to change without advance notice.

Height. Right Prism. Dates, assignments, and quizzes subject to change without advance notice. Name: Period GL UNIT 11: SOLIDS I can define, identify and illustrate the following terms: Face Isometric View Net Edge Polyhedron Volume Vertex Cylinder Hemisphere Cone Cross section Height Pyramid Prism

More information

Geometry Chapter 12. Volume. Surface Area. Similar shapes ratio area & volume

Geometry Chapter 12. Volume. Surface Area. Similar shapes ratio area & volume Geometry Chapter 12 Volume Surface Area Similar shapes ratio area & volume Date Due Section Topics Assignment Written Exercises 12.1 Prisms Altitude Lateral Faces/Edges Right vs. Oblique Cylinders 12.3

More information

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures.

PERIMETER AND AREA. In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. PERIMETER AND AREA In this unit, we will develop and apply the formulas for the perimeter and area of various two-dimensional figures. Perimeter Perimeter The perimeter of a polygon, denoted by P, is the

More information

17.1 Cross Sections and Solids of Rotation

17.1 Cross Sections and Solids of Rotation Name Class Date 17.1 Cross Sections and Solids of Rotation Essential Question: What tools can you use to visualize solid figures accurately? Explore G.10.A Identify the shapes of two-dimensional cross-sections

More information

Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams:

Area of a triangle: The area of a triangle can be found with the following formula: You can see why this works with the following diagrams: Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 You can see why this works with the following diagrams: h h b b Solve: Find the area of

More information

Geometry Vocabulary Booklet

Geometry Vocabulary Booklet Geometry Vocabulary Booklet Geometry Vocabulary Word Everyday Expression Example Acute An angle less than 90 degrees. Adjacent Lying next to each other. Array Numbers, letter or shapes arranged in a rectangular

More information

ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone.

ACTIVITY: Finding a Formula Experimentally. Work with a partner. Use a paper cup that is shaped like a cone. 8. Volumes of Cones How can you find the volume of a cone? You already know how the volume of a pyramid relates to the volume of a prism. In this activity, you will discover how the volume of a cone relates

More information

12-4 Volumes of Prisms and Cylinders. Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h

12-4 Volumes of Prisms and Cylinders. Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h Find the volume of each prism. The volume V of a prism is V = Bh, where B is the area of a base and h The volume is 108 cm 3. The volume V of a prism is V = Bh, where B is the area of a base and h the

More information

Name: Date: Geometry Solid Geometry. Name: Teacher: Pd:

Name: Date: Geometry Solid Geometry. Name: Teacher: Pd: Name: Date: Geometry 2012-2013 Solid Geometry Name: Teacher: Pd: Table of Contents DAY 1: SWBAT: Calculate the Volume of Prisms and Cylinders Pgs: 1-7 HW: Pgs: 8-10 DAY 2: SWBAT: Calculate the Volume of

More information

Geometry and Measurement

Geometry and Measurement The student will be able to: Geometry and Measurement 1. Demonstrate an understanding of the principles of geometry and measurement and operations using measurements Use the US system of measurement for

More information

Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area. Determine the area of various shapes Circumference

Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area. Determine the area of various shapes Circumference 1 P a g e Grade 9 Mathematics Unit 3: Shape and Space Sub Unit #1: Surface Area Lesson Topic I Can 1 Area, Perimeter, and Determine the area of various shapes Circumference Determine the perimeter of various

More information

Surface Area and Volume

Surface Area and Volume UNIT 7 Surface Area and Volume Managers of companies that produce food products must decide how to package their goods, which is not as simple as you might think. Many factors play into the decision of

More information

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units

(a) 5 square units. (b) 12 square units. (c) 5 3 square units. 3 square units. (d) 6. (e) 16 square units 1. Find the area of parallelogram ACD shown below if the measures of segments A, C, and DE are 6 units, 2 units, and 1 unit respectively and AED is a right angle. (a) 5 square units (b) 12 square units

More information

Areas of Rectangles and Parallelograms

Areas of Rectangles and Parallelograms CONDENSED LESSON 8.1 Areas of Rectangles and Parallelograms In this lesson you will Review the formula for the area of a rectangle Use the area formula for rectangles to find areas of other shapes Discover

More information

12-4 Volumes of Prisms and Cylinders. Find the volume of each prism.

12-4 Volumes of Prisms and Cylinders. Find the volume of each prism. Find the volume of each prism. 3. the oblique rectangular prism shown at the right 1. The volume V of a prism is V = Bh, where B is the area of a base and h is the height of the prism. If two solids have

More information

Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples:

Platonic Solids. Some solids have curved surfaces or a mix of curved and flat surfaces (so they aren't polyhedra). Examples: Solid Geometry Solid Geometry is the geometry of three-dimensional space, the kind of space we live in. Three Dimensions It is called three-dimensional or 3D because there are three dimensions: width,

More information

11-1. Space Figures and Cross Sections. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary

11-1. Space Figures and Cross Sections. Vocabulary. Review. Vocabulary Builder. Use Your Vocabulary 11-1 Space Figures and Cross Sections Vocabulary Review Complete each statement with the correct word from the list. edge edges vertex vertices 1. A(n) 9 is a segment that is formed by the intersections

More information

Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in

Area of a triangle: The area of a triangle can be found with the following formula: 1. 2. 3. 12in Area Review Area of a triangle: The area of a triangle can be found with the following formula: 1 A 2 bh or A bh 2 Solve: Find the area of each triangle. 1. 2. 3. 5in4in 11in 12in 9in 21in 14in 19in 13in

More information

Begin recognition in EYFS Age related expectation at Y1 (secure use of language)

Begin recognition in EYFS Age related expectation at Y1 (secure use of language) For more information - http://www.mathsisfun.com/geometry Begin recognition in EYFS Age related expectation at Y1 (secure use of language) shape, flat, curved, straight, round, hollow, solid, vertexvertices

More information

Solids. Objective A: Volume of a Solids

Solids. Objective A: Volume of a Solids Solids Math00 Objective A: Volume of a Solids Geometric solids are figures in space. Five common geometric solids are the rectangular solid, the sphere, the cylinder, the cone and the pyramid. A rectangular

More information

Geometry Notes PERIMETER AND AREA

Geometry Notes PERIMETER AND AREA Perimeter and Area Page 1 of 57 PERIMETER AND AREA Objectives: After completing this section, you should be able to do the following: Calculate the area of given geometric figures. Calculate the perimeter

More information

In Problems #1 - #4, find the surface area and volume of each prism.

In Problems #1 - #4, find the surface area and volume of each prism. Geometry Unit Seven: Surface Area & Volume, Practice In Problems #1 - #4, find the surface area and volume of each prism. 1. CUBE. RECTANGULAR PRISM 9 cm 5 mm 11 mm mm 9 cm 9 cm. TRIANGULAR PRISM 4. TRIANGULAR

More information

Surface Area and Volume Nets to Prisms

Surface Area and Volume Nets to Prisms Surface Area and Volume Nets to Prisms Michael Fauteux Rosamaria Zapata CK12 Editor Say Thanks to the Authors Click http://www.ck12.org/saythanks (No sign in required) To access a customizable version

More information

Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min.

Teacher Page Key. Geometry / Day # 13 Composite Figures 45 Min. Teacher Page Key Geometry / Day # 13 Composite Figures 45 Min. 9-1.G.1. Find the area and perimeter of a geometric figure composed of a combination of two or more rectangles, triangles, and/or semicircles

More information

Session 9 Solids. congruent regular polygon vertex. cross section edge face net Platonic solid polyhedron

Session 9 Solids. congruent regular polygon vertex. cross section edge face net Platonic solid polyhedron Key Terms for This Session Session 9 Solids Previously Introduced congruent regular polygon vertex New in This Session cross section edge face net Platonic solid polyhedron Introduction In this session,

More information

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points.

56 questions (multiple choice, check all that apply, and fill in the blank) The exam is worth 224 points. 6.1.1 Review: Semester Review Study Sheet Geometry Core Sem 2 (S2495808) Semester Exam Preparation Look back at the unit quizzes and diagnostics. Use the unit quizzes and diagnostics to determine which

More information

12-1 Representations of Three-Dimensional Figures

12-1 Representations of Three-Dimensional Figures Connect the dots on the isometric dot paper to represent the edges of the solid. Shade the tops of 12-1 Representations of Three-Dimensional Figures Use isometric dot paper to sketch each prism. 1. triangular

More information

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry

TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry TargetStrategies Aligned Mathematics Strategies Arkansas Student Learning Expectations Geometry ASLE Expectation: Focus Objective: Level: Strand: AR04MGE040408 R.4.G.8 Analyze characteristics and properties

More information

Name Date Period. 3D Geometry Project

Name Date Period. 3D Geometry Project Name 3D Geometry Project Part I: Exploring Three-Dimensional Shapes In the first part of this WebQuest, you will be exploring what three-dimensional (3D) objects are, how to classify them, and several

More information

MENSURATION. Definition

MENSURATION. Definition MENSURATION Definition 1. Mensuration : It is a branch of mathematics which deals with the lengths of lines, areas of surfaces and volumes of solids. 2. Plane Mensuration : It deals with the sides, perimeters

More information

Grade 7/8 Math Circles Winter D Geometry

Grade 7/8 Math Circles Winter D Geometry 1 University of Waterloo Faculty of Mathematics Grade 7/8 Math Circles Winter 2013 3D Geometry Introductory Problem Mary s mom bought a box of 60 cookies for Mary to bring to school. Mary decides to bring

More information

FS Geometry EOC Review

FS Geometry EOC Review MAFS.912.G-C.1.1 Dilation of a Line: Center on the Line In the figure, points A, B, and C are collinear. http://www.cpalms.org/public/previewresource/preview/72776 1. Graph the images of points A, B, and

More information

Grade 7/8 Math Circles Winter D Geometry

Grade 7/8 Math Circles Winter D Geometry 1 University of Waterloo Faculty of Mathematics Grade 7/8 Math Circles Winter 2013 3D Geometry Introductory Problem Mary s mom bought a box of 60 cookies for Mary to bring to school. Mary decides to bring

More information

Chapter 8. Chapter 8 Opener. Section 8.1. Big Ideas Math Green Worked-Out Solutions. Try It Yourself (p. 353) Number of cubes: 7

Chapter 8. Chapter 8 Opener. Section 8.1. Big Ideas Math Green Worked-Out Solutions. Try It Yourself (p. 353) Number of cubes: 7 Chapter 8 Opener Try It Yourself (p. 5). The figure is a square.. The figure is a rectangle.. The figure is a trapezoid. g. Number cubes: 7. a. Sample answer: 4. There are 5 6 0 unit cubes in each layer.

More information

Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg

Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg Surface Area and Volume Learn it, Live it, and Apply it! Elizabeth Denenberg Objectives My unit for the Delaware Teacher s Institute is a 10 th grade mathematics unit that focuses on teaching students

More information

Geometry Notes VOLUME AND SURFACE AREA

Geometry Notes VOLUME AND SURFACE AREA Volume and Surface Area Page 1 of 19 VOLUME AND SURFACE AREA Objectives: After completing this section, you should be able to do the following: Calculate the volume of given geometric figures. Calculate

More information

Area is a measure of how much space is occupied by a figure. 1cm 1cm

Area is a measure of how much space is occupied by a figure. 1cm 1cm Area Area is a measure of how much space is occupied by a figure. Area is measured in square units. For example, one square centimeter (cm ) is 1cm wide and 1cm tall. 1cm 1cm A figure s area is the number

More information

Activity Set 4. Trainer Guide

Activity Set 4. Trainer Guide Geometry and Measurement of Solid Figures Activity Set 4 Trainer Guide Mid_SGe_04_TG Copyright by the McGraw-Hill Companies McGraw-Hill Professional Development GEOMETRY AND MEASUREMENT OF SOLID FIGURES

More information

Geo - CH10 Practice Test

Geo - CH10 Practice Test Geo - H10 Practice Test Multiple hoice Identify the choice that best completes the statement or answers the question. 1. lassify the figure. Name the vertices, edges, and base. a. triangular pyramid vertices:,,,,

More information

Chapter 4: Area, Perimeter, and Volume. Geometry Assessments

Chapter 4: Area, Perimeter, and Volume. Geometry Assessments Chapter 4: Area, Perimeter, and Volume Geometry Assessments Area, Perimeter, and Volume Introduction The performance tasks in this chapter focus on applying the properties of triangles and polygons to

More information

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line.

Geometry Chapter 1 Vocabulary. coordinate - The real number that corresponds to a point on a line. Chapter 1 Vocabulary coordinate - The real number that corresponds to a point on a line. point - Has no dimension. It is usually represented by a small dot. bisect - To divide into two congruent parts.

More information

II. Geometry and Measurement

II. Geometry and Measurement II. Geometry and Measurement The Praxis II Middle School Content Examination emphasizes your ability to apply mathematical procedures and algorithms to solve a variety of problems that span multiple mathematics

More information

MATH 139 FINAL EXAM REVIEW PROBLEMS

MATH 139 FINAL EXAM REVIEW PROBLEMS MTH 139 FINL EXM REVIEW PROLEMS ring a protractor, compass and ruler. Note: This is NOT a practice exam. It is a collection of problems to help you review some of the material for the exam and to practice

More information

3D Geometry: Chapter Questions

3D Geometry: Chapter Questions 3D Geometry: Chapter Questions 1. What are the similarities and differences between prisms and pyramids? 2. How are polyhedrons named? 3. How do you find the cross-section of 3-Dimensional figures? 4.

More information

PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT

PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES NCERT UNIT 12 PRACTICAL GEOMETRY SYMMETRY AND VISUALISING SOLID SHAPES (A) Main Concepts and Results Let a line l and a point P not lying on it be given. By using properties of a transversal and parallel lines,

More information

17.2 Surface Area of Prisms and Cylinders

17.2 Surface Area of Prisms and Cylinders Name Class Date 17. Surface Area of Prisms and Cylinders Essential Question: How can you find the surface area of a prism or cylinder? Explore G.11.C Apply the formulas for the total and lateral surface

More information

Grade 4 Math Expressions Vocabulary Words

Grade 4 Math Expressions Vocabulary Words Grade 4 Math Expressions Vocabulary Words Link to Math Expression Online Glossary for some definitions: http://wwwk6.thinkcentral.com/content/hsp/math/hspmathmx/na/gr4/se_9780547153131_/eglos sary/eg_popup.html?grade=4

More information

Practice: Space Figures and Cross Sections Geometry 11-1

Practice: Space Figures and Cross Sections Geometry 11-1 Practice: Space Figures and Cross Sections Geometry 11-1 Name: Date: Period: Polyhedron * 3D figure whose surfaces are * each polygon is a. * an is a segment where two faces intersect. * a is a point where

More information

Su.a Supported: Identify Determine if polygons. polygons with all sides have all sides and. and angles equal angles equal (regular)

Su.a Supported: Identify Determine if polygons. polygons with all sides have all sides and. and angles equal angles equal (regular) MA.912.G.2 Geometry: Standard 2: Polygons - Students identify and describe polygons (triangles, quadrilaterals, pentagons, hexagons, etc.), using terms such as regular, convex, and concave. They find measures

More information

12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017.

12-6 Surface Area and Volumes of Spheres. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. SOLUTION: ANSWER: 1017. Find the surface area of each sphere or hemisphere. Round to the nearest tenth. 3. sphere: area of great circle = 36Ļ€ yd 2 We know that the area of a great circle is r.. Find 1. Now find the surface area.

More information

Geometry Concepts. Figures that lie in a plane are called plane figures. These are all plane figures. Triangle 3

Geometry Concepts. Figures that lie in a plane are called plane figures. These are all plane figures. Triangle 3 Geometry Concepts Figures that lie in a plane are called plane figures. These are all plane figures. Polygon No. of Sides Drawing Triangle 3 A polygon is a plane closed figure determined by three or more

More information

*1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles.

*1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Students: 1. Students understand and compute volumes and areas of simple objects. *1. Derive formulas for the area of right triangles and parallelograms by comparing with the area of rectangles. Review

More information

12-2 Surface Areas of Prisms and Cylinders. 1. Find the lateral area of the prism. SOLUTION: ANSWER: in 2

12-2 Surface Areas of Prisms and Cylinders. 1. Find the lateral area of the prism. SOLUTION: ANSWER: in 2 1. Find the lateral area of the prism. 3. The base of the prism is a right triangle with the legs 8 ft and 6 ft long. Use the Pythagorean Theorem to find the length of the hypotenuse of the base. 112.5

More information

Right Prisms Let s find the surface area of the right prism given in Figure 44.1. Figure 44.1

Right Prisms Let s find the surface area of the right prism given in Figure 44.1. Figure 44.1 44 Surface Area The surface area of a space figure is the total area of all the faces of the figure. In this section, we discuss the surface areas of some of the space figures introduced in Section 41.

More information

Algebra Geometry Glossary. 90 angle

Algebra Geometry Glossary. 90 angle lgebra Geometry Glossary 1) acute angle an angle less than 90 acute angle 90 angle 2) acute triangle a triangle where all angles are less than 90 3) adjacent angles angles that share a common leg Example:

More information

Grade 11 Essential Mathematics Unit 6: Measurement and Geometry

Grade 11 Essential Mathematics Unit 6: Measurement and Geometry Grade 11 Essential Mathematics Unit 6: INTRODUCTION When people first began to take measurements, they would use parts of the hands and arms. For example, a digit was the width of a thumb. This kind of

More information

Sixth Grade Math Pacing Guide Page County Public Schools MATH 6/7 1st Nine Weeks: Days Unit: Decimals B

Sixth Grade Math Pacing Guide Page County Public Schools MATH 6/7 1st Nine Weeks: Days Unit: Decimals B Sixth Grade Math Pacing Guide MATH 6/7 1 st Nine Weeks: Unit: Decimals 6.4 Compare and order whole numbers and decimals using concrete materials, drawings, pictures and mathematical symbols. 6.6B Find

More information

Perimeter and Area. An artist uses perimeter and area to determine the amount of materials it takes to produce a piece such as this.

Perimeter and Area. An artist uses perimeter and area to determine the amount of materials it takes to produce a piece such as this. UNIT 10 Perimeter and Area An artist uses perimeter and area to determine the amount of materials it takes to produce a piece such as this. 3 UNIT 10 PERIMETER AND AREA You can find geometric shapes in

More information

Measuring Prisms and Cylinders. Suggested Time: 5 Weeks

Measuring Prisms and Cylinders. Suggested Time: 5 Weeks Measuring Prisms and Cylinders Suggested Time: 5 Weeks Unit Overview Focus and Context In this unit, students will use two-dimensional nets to create threedimensional solids. They will begin to calculate

More information

GAP CLOSING. Volume and Surface Area. Intermediate / Senior Student Book

GAP CLOSING. Volume and Surface Area. Intermediate / Senior Student Book GAP CLOSING Volume and Surface Area Intermediate / Senior Student Book Volume and Surface Area Diagnostic...3 Volumes of Prisms...6 Volumes of Cylinders...13 Surface Areas of Prisms and Cylinders...18

More information

2 feet Opposite sides of a rectangle are equal. All sides of a square are equal. 2 X 3 = 6 meters = 18 meters

2 feet Opposite sides of a rectangle are equal. All sides of a square are equal. 2 X 3 = 6 meters = 18 meters GEOMETRY Vocabulary 1. Adjacent: Next to each other. Side by side. 2. Angle: A figure formed by two straight line sides that have a common end point. A. Acute angle: Angle that is less than 90 degree.

More information

SOLIDS, NETS, AND CROSS SECTIONS

SOLIDS, NETS, AND CROSS SECTIONS SOLIDS, NETS, AND CROSS SECTIONS Polyhedra In this section, we will examine various three-dimensional figures, known as solids. We begin with a discussion of polyhedra. Polyhedron A polyhedron is a three-dimensional

More information

Archdiocese of Washington Catholic Schools Academic Standards Mathematics

Archdiocese of Washington Catholic Schools Academic Standards Mathematics 5 th GRADE Archdiocese of Washington Catholic Schools Standard 1 - Number Sense Students compute with whole numbers*, decimals, and fractions and understand the relationship among decimals, fractions,

More information

The Area is the width times the height: Area = w h

The Area is the width times the height: Area = w h Geometry Handout Rectangle and Square Area of a Rectangle and Square (square has all sides equal) The Area is the width times the height: Area = w h Example: A rectangle is 6 m wide and 3 m high; what

More information

B = 1 14 12 = 84 in2. Since h = 20 in then the total volume is. V = 84 20 = 1680 in 3

B = 1 14 12 = 84 in2. Since h = 20 in then the total volume is. V = 84 20 = 1680 in 3 45 Volume Surface area measures the area of the two-dimensional boundary of a threedimensional figure; it is the area of the outside surface of a solid. Volume, on the other hand, is a measure of the space

More information

*1. Understand the concept of a constant number like pi. Know the formula for the circumference and area of a circle.

*1. Understand the concept of a constant number like pi. Know the formula for the circumference and area of a circle. Students: 1. Students deepen their understanding of measurement of plane and solid shapes and use this understanding to solve problems. *1. Understand the concept of a constant number like pi. Know the

More information

Mensuration Introduction

Mensuration Introduction Mensuration Introduction Mensuration is the process of measuring and calculating with measurements. Mensuration deals with the determination of length, area, or volume Measurement Types The basic measurement

More information

10-4 Surface Area of Prisms and Cylinders

10-4 Surface Area of Prisms and Cylinders : Finding Lateral Areas and Surface Areas of Prisms 2. Find the lateral area and surface area of the right rectangular prism. : Finding Lateral Areas and Surface Areas of Right Cylinders 3. Find the lateral

More information

Centroid: The point of intersection of the three medians of a triangle. Centroid

Centroid: The point of intersection of the three medians of a triangle. Centroid Vocabulary Words Acute Triangles: A triangle with all acute angles. Examples 80 50 50 Angle: A figure formed by two noncollinear rays that have a common endpoint and are not opposite rays. Angle Bisector:

More information

12-8 Congruent and Similar Solids

12-8 Congruent and Similar Solids Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. 3. Two similar cylinders have radii of 15 inches and 6 inches. What is the ratio

More information

FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST. Mathematics Reference Sheets. Copyright Statement for this Assessment and Evaluation Services Publication

FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST. Mathematics Reference Sheets. Copyright Statement for this Assessment and Evaluation Services Publication FCAT FLORIDA COMPREHENSIVE ASSESSMENT TEST Mathematics Reference Sheets Copyright Statement for this Assessment and Evaluation Services Publication Authorization for reproduction of this document is hereby

More information

Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry.

Geometry. Geometry is the study of shapes and sizes. The next few pages will review some basic geometry facts. Enjoy the short lesson on geometry. Geometry Introduction: We live in a world of shapes and figures. Objects around us have length, width and height. They also occupy space. On the job, many times people make decision about what they know

More information

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees

Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Angle - a figure formed by two rays or two line segments with a common endpoint called the vertex of the angle; angles are measured in degrees Apex in a pyramid or cone, the vertex opposite the base; in

More information

Volume of Pyramids and Cones. Tape together as shown. Tape together as shown.

Volume of Pyramids and Cones. Tape together as shown. Tape together as shown. 7-6 Volume of Pyramids and Cones MAIN IDEA Find the volumes of pyramids and cones. New Vocabulary cone Math Online glencoe.com Extra Examples Personal Tutor Self-Check Quiz In this Mini Lab, you will investigate

More information

VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region.

VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. Math 6 NOTES 7.5 Name VOLUME of Rectangular Prisms Volume is the measure of occupied by a solid region. **The formula for the volume of a rectangular prism is:** l = length w = width h = height Study Tip:

More information

Angles that are between parallel lines, but on opposite sides of a transversal.

Angles that are between parallel lines, but on opposite sides of a transversal. GLOSSARY Appendix A Appendix A: Glossary Acute Angle An angle that measures less than 90. Acute Triangle Alternate Angles A triangle that has three acute angles. Angles that are between parallel lines,

More information

Most classrooms are built in the shape of a rectangular prism. You will probably find yourself inside a polyhedron at school!

Most classrooms are built in the shape of a rectangular prism. You will probably find yourself inside a polyhedron at school! 3 D OBJECTS Properties of 3 D Objects A 3 Dimensional object (3 D) is a solid object that has 3 dimensions, i.e. length, width and height. They take up space. For example, a box has three dimensions, i.e.

More information

12-8 Congruent and Similar Solids

12-8 Congruent and Similar Solids Determine whether each pair of solids is similar, congruent, or neither. If the solids are similar, state the scale factor. Ratio of radii: Ratio of heights: The ratios of the corresponding measures are

More information

, where B is the area of the base and h is the height of the pyramid. The base

, where B is the area of the base and h is the height of the pyramid. The base Find the volume of each pyramid. The volume of a pyramid is, where B is the area of the base and h is the height of the pyramid. The base of this pyramid is a right triangle with legs of 9 inches and 5

More information

Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square.

Show that when a circle is inscribed inside a square the diameter of the circle is the same length as the side of the square. Week & Day Week 6 Day 1 Concept/Skill Perimeter of a square when given the radius of an inscribed circle Standard 7.MG:2.1 Use formulas routinely for finding the perimeter and area of basic twodimensional

More information

An arrangement that shows objects in rows and columns Example:

An arrangement that shows objects in rows and columns Example: 1. acute angle : An angle that measures less than a right angle (90 ). 2. addend : Any of the numbers that are added 2 + 3 = 5 The addends are 2 and 3. 3. angle : A figure formed by two rays that meet

More information

Calculating the surface area of a three-dimensional object is similar to finding the area of a two dimensional object.

Calculating the surface area of a three-dimensional object is similar to finding the area of a two dimensional object. Calculating the surface area of a three-dimensional object is similar to finding the area of a two dimensional object. Surface area is the sum of areas of all the faces or sides of a three-dimensional

More information

11-4 Areas of Regular Polygons and Composite Figures

11-4 Areas of Regular Polygons and Composite Figures 1. In the figure, square ABDC is inscribed in F. Identify the center, a radius, an apothem, and a central angle of the polygon. Then find the measure of a central angle. Center: point F, radius:, apothem:,

More information