1. Three-Dimensional Figures Volume and Surface Area

2. Warm Up Identify each two-dimensional figure described. 1. four sides that are all congruent 2. six sides 3. four sides with parallel opposite sides 4. four right angles and four congruent sides rhombus hexagon parallelogram square

3. Vocabulary face edge polyhedron vertex base prism pyramid cylinder cone

4. Three-dimensional figures have three- dimensions: length, width, and height. A flat surface of a three-dimensional figure is a face. An edge is where two faces meet. A polyhedron is a three-dimensional figure whose faces are all polygons. A vertex of a polyhedron is a point where three or more edges meet. The face that is used to name a polyhedron is called a base.

6. Additional Example 1: Naming Prisms and Pyramids There is one base, and it is a pentagon. There are five triangular faces. The figure is a pentagonal pyramid. Identify the bases and faces of the figure. Then name the figure. A.

7. Additional Example 1: Naming Prisms and Pyramids There is one base, and it is a triangle. There are three triangular faces. The figure is a triangular pyramid. Describe the bases and faces of the figure. Then name the figure. B.

8. Additional Example 1: Naming Prisms and Pyramids There are two bases, and they are both hexagons. There are six rectangular faces. The figure is a hexagonal prism. Describe the bases and faces of the figure. Then name the figure. C.

9. Check It Out! Example 1 There are two bases and they are both triangles. There are three rectangular faces. The figure is a triangular prism. Describe the bases and faces of the figure. Then name the figure. A.

10. Check It Out! Example 1 There are two rectangular bases. The figure is a rectangular prism. Describe the bases and faces of the figure. Then name the figure. B. There are four other rectangular faces.

11. Describe the bases and faces of the figure. Then name the figure. C. Check It Out! Example 1 There are two octagonal bases. The figure is an octagonal prism. There are eight rectangular faces.

12. Other three-dimensional figures include cylinders and cones. These figures are not polyhedrons because they are not made of faces that are all polygons.

13. You can use properties to classify three- dimensional figures.

14. Classify each figure as a polyhedron or not a polyhedron. Then name the figure. A. Additional Example 2: Classifying Three- Dimensional Figures The faces are all polygons, so the figure is a polyhedron. There is one rectangular base for each figure. The figure is made up of a rectangular pyramid and a rectangular prism.

15. There is one circular base. Classify each figure as a polyhedron or not a polyhedron. Then name the figure. B. Additional Example 2: Classifying Three- Dimensional Figures The faces are not all polygons, so the figure is not a polyhedron. The figure is a cone.

16. There are two circular bases. Classify each figure as a polyhedron or not a polyhedron. Then name the figure. C. Additional Example 2: Classifying Three- Dimensional Figures The faces are not all polygons, so the figure is not a polyhedron. The figure is a cylinder.

17. Check It Out! Example 2 Classify each figure as a polyhedron or not a polyhedron. Then name the figure. A. There is one circular base for the top figure and two circular bases for the bottom figure. The faces are not all polygons, so the figure is not a polyhedron. The figure is made up of a cylinder and a cone.

18. Check It Out! Example 2 Classify each figure as a polyhedron or not a polyhedron. Then name the figure. B. There are two triangular bases for the figure. The faces are all polygons, so the figure is a polyhedron. The figure is a triangular prism.

19. Check It Out! Example 2 Classify each figure as a polyhedron or not a polyhedron. Then name the figure. C. There is one square base for the figure. The faces are all polygons, so the figure is a polyhedron. The figure is a square pyramid.

20. Lesson Quiz: Part I Describe the bases and faces of each figure. Then name each figure. Two pentagonal bases, 5 rectangular faces; pentagonal prism One square base, 4 triangular faces; square pyramid 1. 2.

21. Lesson Quiz: Part II Classify each figure as a polyhedron or not a polyhedron. Then name the figure. polyhedron, rectangular prism polyhedron, triangular prism 3. 4.

22. Warm Up Find the area of each figure described. Use 3.14 for . 1. a triangle with a base of 6 feet and a height of 3 feet 2. a circle with radius 5 in. 9 ft 2 78.5 in 2

23. Volume

24. The volume of a three-dimensional figure is the number of cubes it can hold. Each cube represents a unit of measure called a cubic unit.

25. Height Triangular prism Rectangular prism Cylinder Base Height Base Height Base

27. Find the volume of each figure to the nearest tenth. A. Additional Example 1: Finding the Volume of Prisms and Cylinders = 192 ft 3 B = 4 12 = 48 ft 2 V = Bh = 48 4 The base is a rectangle. Volume of a prism Substitute for B and h. Multiply.

28. Find the volume of the figure to the nearest tenth. Use 3.14 for . B. = 192  602.9 in 3 B =  (4 2 ) = 16  in 2 V = Bh = 16  12 Additional Example 1: Finding the Volume of Prisms and Cylinders The base is a circle. Volume of a cylinder Substitute for B and h. Multiply.

29. Find the volume of the figure to the nearest tenth. Use 3.14 for . C. 7 ft V = Bh = 15 7 = 105 ft 3 B = 6 5 = 15 ft 2 1212 Additional Example 1: Finding the Volume of Prisms and Cylinders The base is a triangle. Volume of a prism Substitute for B and h. Multiply.

30. Find the volume of the figure to the nearest tenth. Use 3.14 for . A. = 180 in 3 B = 6 3 = 18 in. 2 V = Bh = 18 10 The base is a rectangle. Volume of prism Check It Out! Example 1 Substitute for B and h. Multiply. 10 in. 6 in. 3 in.

31. Find the volume of the figure to the nearest tenth. Use 3.14 for . B. 8 cm 15 cm B =  (8 2 ) = 64  cm 2 = (64  )(15) = 960   3,014.4 cm 3 Check It Out! Example 1 The base is a circle. Volume of a cylinder V = Bh Substitute for B and h. Multiply.

32. Find the volume of the figure to the nearest tenth. C. 10 ft 14 ft 12 ft = 60 ft 2 = 60(14) = 840 ft 3 Check It Out! Example 1 The base is a triangle. Volume of a prism B = 12 10 1212 V = Bh Substitute for B and h. Multiply.

33. A juice box measures 3 in. by 2 in. by 4 in. Explain whether tripling only the length, width, or height of the box would triple the amount of juice the box holds. Additional Example 2A: Exploring the Effects of Changing Dimensions The original box has a volume of 24 in 3. You could triple the volume to 72 in 3 by tripling any one of the dimensions. So tripling the length, width, or height would triple the amount of juice the box holds.

34. A juice can has a radius of 2 in. and a height of 5 in. Explain whether tripling only the height of the can would have the same effect on the volume as tripling the radius. Additional Example 2B: Exploring the Effects of Changing Dimensions By tripling the height, you would triple the volume. By tripling the radius, you would increase the volume to 9 times the original volume.

35. A box measures 5 in. by 3 in. by 7 in. Explain whether tripling only the length, width, or height of the box would triple the volume of the box. Check It Out! Example 2A Tripling the length would triple the volume. V = (15)(3)(7) = 315 cm 3 The original box has a volume of (5)(3)(7) = 105 cm 3.

36. Check It Out! Example 2A Continued The original box has a volume of (5)(3)(7) = 105 cm 3. Tripling the height would triple the volume. V = (5)(3)(21) = 315 cm 3

37. Check It Out! Example 2A Continued Tripling the width would triple the volume. V = (5)(9)(7) = 315 cm 3 The original box has a volume of (5)(3)(7) = 105 cm 3.

38. By tripling the radius, you would increase the volume nine times. A cylinder measures 3 cm tall with a radius of 2 cm. Explain whether tripling only the radius or height of the cylinder would triple the amount of volume. Check It Out! Example 2B V = 36  3 = 108  cm 3 The original cylinder has a volume of 4  3 = 12  cm 3.

39. Check It Out! Example 2B Continued Tripling the height would triple the volume. V = 4  9 = 36  cm 3 The original cylinder has a volume of 4  3 = 12  cm 3.

40. A drum company advertises a snare drum that is 4 inches high and 12 inches in diameter. Estimate the volume of the drum. Additional Example 3: Music Application d = 12, h = 4 r = = = 6 Volume of a cylinder d2d2 V = (  r 2 )h 12 2 = (3.14)(6) 2 4 = (3.14)(36)(4) = 452.16 ≈ 452 Use 3.14 for . The volume of the drum is approximately 452 in 3.

41. A drum company advertises a bass drum that is 12 inches high and 28 inches in diameter. Estimate the volume of the drum. Check It Out! Example 3 d = 28, h = 12 r = = = 14 Volume of a cylinder d2d2 V = (  r 2 )h 28 2 = (3.14)(14) 2 12 = (3.14)(196)(12) = 7385.28 ≈ 7,385 Use 3.14 for . The volume of the drum is approximately 7,385 in 3.

42. Find the volume of the the barn. Volume of barn Volume of rectangular prism Volume of triangular prism + = = 30,000 + 10,000 V = (40)(50)(15) + (40)(10)(50) 1212 = 40,000 ft 3 The volume of the barn is 40,000 ft 3. Additional Example 4: Finding the Volume of Composite Figures

43. Check It Out! Example 4 Find the volume of the play house. 3 ft 4 ft 8 ft 5 ft V = (8)(3)(4) + (5)(8)(3) 1212 = 96 + 60 V = 156 ft 3 Volume of house Volume of rectangular prism Volume of triangular prism + = The volume of the play house is 156 ft 3.

44. Lesson Quiz Find the volume of each figure to the nearest tenth. Use 3.14 for . 306 in 3 942 in 3 160.5 in 3 No; the volume would be quadrupled because you have to use the square of the radius to find the volume. 10 in. 8.5 in. 3 in. 12 in. 2 in. 15 in. 10.7 in. 1.3. 2. 4. Explain whether doubling the radius of the cylinder above will double the volume.

45. The height of a pyramid or cone is measured from the highest point to the base along a perpendicular line.

47. Additional Example 1: Finding the Volume of Pyramids and Cones Find the volume of the figure. A. 1313 V = 14 6 V = 28 cm 3 V = Bh 1313 B = (4 7) = 14 cm 2 1212

48. Additional Example 1: Finding the Volume of Pyramids and Cones 1313 V = 9 10 V = 30  94.2 in 3 V = Bh 1313 B = (3 2 ) = 9 in 2 Use 3.14 for . Find the volume of the figure. Use 3.14 for . B.

49. Check It Out! Example 1 1313 V = 17.5 7 V  40.8 in 3 V = Bh 1313 B = (5 7) = 17.5 in 2 1212 5 in. 7 in. Find the volume of the figure. A.

50. 1313 V = 9 7 V = 21  65.9 m 3 V = Bh 1313 B = (3 2 ) = 9 m 2 Use 3.14 for . Check It Out! Example 1 7 m 3 m Find the volume of the figure. Use 3.14 for . B.

51. Additional Example 2: Social Studies Application The Pyramid of Kukulcán in Mexico is a square pyramid. Its height is 24 m and its base has 55 m sides. Find the volume of the pyramid. B = 55 2 The base is a square. Step 1: Find the area of the base. = 3025 m 2 Multiply.

52. Additional Example 2 Continued The Pyramid of Kukulcán in Mexico is a square pyramid. Its height is 24 m and its base has 55 m sides. Find the volume of the pyramid. 1313 V = (3025)(24) V = 24,200 m 3 The volume of the pyramid is 24,200 m 3. Step 2: Find the volume. 1313 V = Bh Write the formula. Substitute for B and h. Multiply.

53. Check It Out! Example 2 B = 48 2 The base is a square. Step 1: Find the area of the base. = 2304 m 2 Multiply. Find the volume of a pyramid with a height of 12 m and a base with 48 m sides.

54. Check It Out! Example 2 Continued 1313 V = (2304)(12) V ≈ 9216 m 3 The volume of the pyramid is 9,216 m 3. Step 2: Find the volume. 1313 V = Bh Write the formula. Substitute for B and h. Multiply. Find the volume of a pyramid with a height of 12 m and a base with 48 m sides.

55. Lesson Quiz: Part l Find the volume of each figure to the nearest tenth. Use 3.14 for . 78.5 in 3 6.3 m 3 1. the triangular pyramid 2. the cone

56. Lesson Quiz: Part ll Yes; the volume is one-third the product of the base area and the height. So if you triple the height, the product would be tripled. 3. Explain whether tripling the height of the triangular pyramid would triple the volume.

57. Three-Dimensional Figures Surface Area

58. Warm Up 1. A triangular pyramid has a base area of 1.2 m 2 and a height of 7.5 m. What is the volume of the pyramid? 2. A cone has a radius of 4 cm and a height of 10 cm. What is the volume of the cone to the nearest cubic centimeter? Use 3.14 for . 3 m 3 167 cm 3

59. Vocabulary surface area lateral face lateral area lateral surface

60. The surface area of a three-dimensional figure is the sum of the areas of all its surfaces.

61. Draw each view of the figure. Additional Example 1A: Finding Surface Area of Figures Built of Cubes Find the surface area of each figure. The figure is made up of congruent cubes. topfrontleft bottombackright 1 cm Find the area of each view. 12 + 8 + 6 + 12 + 8 + 6 = 52 The surface area is 52 cm 2.

62. Draw each view of the figure. Additional Example 1B: Finding Surface Area of Figures Built of Cubes Find the surface area of each figure. The figure is made up of congruent cubes. topfrontleft bottombackright 1 cm Find the area of each view. 8 + 8 + 6 + 8 + 8 + 6 = 44 The surface area is 44 cm 2.

63. Draw each view of the figure. Check It Out! Example 1A Find the surface area of each figure. The figure is made up of congruent cubes. topfrontleft bottombackright 1 cm Find the area of each view. 8 + 8 + 4 + 8 + 8 + 4 = 40 The surface area is 40 cm 2.

64. Draw each view of the figure. Check It Out! Example 1B Find the surface area of each figure. The figure is made up of congruent cubes. topfrontleft bottombackright 1 cm Find the area of each view. 8 + 9 + 6 + 8 + 9 + 6 = 46 The surface area is 46 cm 2.

65. The lateral faces of a prism are parallelograms that connect the bases. The lateral area of a prism is the sum of the areas of the lateral faces.

67. S = 2B + Ph = 204 ft 2 = 2( 8 3) + (18)(10) 1212 Additional Example 2: Finding Surface Area of Prisms Find the surface area of the figure to the nearest tenth. The figure is a triangular prism.

68. S = 2B + Ph = 252 cm 2 = 2( 7 6) + (21)(10) 1212 Check It Out! Example 2 7 cm 10 cm 6 cm Find the surface area of the figure to the nearest tenth. The figure is a triangular prism.

69. The lateral surface of a cylinder is the curved surface.

70. S = 2r 2 + 2rh = 2(4 2 ) + 2(4)(6) = 80 in 2  251.2 in 2 Additional Example 3: Finding Surface Area of Cylinders Find the surface area of the cylinder to the nearest tenth. Use 3.14 for .

71. S = 2r 2 + 2rh = 2(15 2 ) + 2(15)(3) = 540 in 2  1695.6 cm 2 Check It Out! Example 3 15 cm 3 cm Find the surface area of the cylinder to the nearest tenth. Use 3.14 for .

72. Additional Example 4: Application A cylindrical soup can is 7.6 cm in diameter and 11.2 cm tall. Estimate the area of the label that covers the side of the can. Only the lateral surface needs to be covered. Diameter ≈ 8 cm, so r ≈ 4 cm. L = 2rh = 2(4)(11) = 88 ≈ 267.3 cm 2 The cylinder’s diameter is about 8 cm, and its height is about 11 cm.

73. Check It Out! Example 4 A cylindrical storage tank that is 6 ft in diameter and 12 ft tall needs to be painted. Estimate the area to be painted. The diameter is 6 ft, so r = 3 ft. S = 2r 2 + 2rh = 2(3 2 ) + 2(3)(12) = 90 ft 2  282.6 ft 2

74. 3. All outer surfaces of a box are covered with gold foil, except the bottom. The box measures 6 in. long, 4 in. wide, and 3 in. high. How much gold foil was used? Lesson Quiz Find the surface area of each figure to the nearest tenth. Use 3.14 for . 1. the triangular prism 2. the cylinder 320.3 in 2 360 cm 2 84 in 2

75. Surface Area Pyramids and Cones

76. Warm Up 1. A rectangular prism is 0.6 m by 0.4 m by 1.0 m. What is the surface area? 2. A cylindrical can has a diameter of 14 cm and a height of 20 cm. What is the surface area to the nearest tenth? Use 3.14 for . 2.48 m 2 1186.9 cm 2

77. Vocabulary slant height regular pyramid right cone

78. The slant height of a pyramid or cone is measured along its lateral surface. In a right cone, a line perpendicular to the base through the vertex passes through the center of the base. The base of a regular pyramid is a regular polygon, and the lateral faces are all congruent. Right cone Regular Pyramid

81. Additional Example 1: Finding Surface Area Find the surface area of the figure to the nearest tenth. Use 3.14 for . = 20.16 ft 2 S = B + Pl 1212 = (2.4 2.4) + (9.6)(3) 1212

82. Check It Out! Example 1 = (3 3) + (12)(5) 1212 B. S =  r 2 +  rl = 39 m 2 =  (7 2 ) +  (7)(18) = 175   549.5 ft 2 5 m 3 m 7 ft 18 ft A. S = B + Pl 1212 Find the surface area of each figure to the nearest tenth. Use 3.14 for .

83. Additional Example 2: Exploring the Effects of Changing Dimensions A cone has diameter 8 in. and slant height 3 in. Explain whether tripling only the slant height would have the same effect on the surface area as tripling only the radius. Use 3.14 for . They would not have the same effect. Tripling the radius would increase the surface area more than tripling the slant height.

84. Check It Out! Example 2 Original Dimensions Triple the Slant HeightTriple the Radius S =  r 2 +  rl =  (4.5) 2 +  (4.5)(2) = 29.25  in 2  91.8 in 2 S =  r 2 +  r(3l) =  (4.5) 2 +  (4.5)(6) = 47.25  in 2  148.4 in 2 S =  r) 2 +  r)l =  (13.5) 2 +  (13.5)(2) = 209.25  in 2  657.0 in 2 A cone has diameter 9 in. and a slant height 2 in. Explain whether tripling only the slant height would have the same effect on the surface area as tripling only the radius. Use the 3.14 for . They would not have the same effect. Tripling the radius would increase the surface area more than tripling the height.

85. Additional Example 3: Application The upper portion of an hourglass is approximately an inverted cone with the given dimensions. What is the lateral surface area of the upper portion of the hourglass? =  (10)(26)  816.8 mm 2 Pythagorean Theorem Lateral surface area L =  rl a 2 + b 2 = l 2 10 2 + 24 2 = l 2 l = 26

86. Check It Out! Example 3 A large road construction cone is almost a full cone. With the given dimensions, what is the lateral surface area of the cone? =  (9)(15)  424.1 in 2 12 in. 9 in. Pythagorean Theorem a 2 + b 2 = l 2 9 2 + 12 2 = l 2 l = 15 Lateral surface area L =  rl

87. Lesson Quiz: Part I Find the surface area of each figure to the nearest tenth. Use 3.14 for . 1. the triangular pyramid 2. the cone 175.8 in 2 6.2 m 2

88. 3. Tell whether doubling the dimensions of a cone will double the surface area. Lesson Quiz: Part II It will more than double the surface area because you square the radius to find the area of the base.

89. Spheres

90. Warm Up 1. Find the surface area of a square pyramid whose base is 3 m on a side and whose slant height is 5 m. 2. Find the surface area of a cone whose base has a radius of 10 in. and whose slant height is 14 in. Use 3.14 for . 39 m 2 753.6 in 2

91. Vocabulary sphere hemisphere

92. A sphere is the set of points in three dimensions that are a fixed distance from a given point, the center. A plane that intersects a sphere through its center divides the sphere into two halves or hemispheres. The edge of a hemisphere is a great circle.

93. The volume of a hemisphere is exactly halfway between the volume of a cone and a cylinder with the same radius r and height equal to r.

95. Additional Example 1: Finding the Volume of a Sphere Find the volume of a sphere with radius 12 cm, both in terms of  and to the nearest tenth. Use 3.14 for . = 2304 cm 3  7,234.6 cm 3 Volume of a sphere Substitute 12 for r. 4343 V = r 3 = (12) 3 4343

96. Check It Out! Example 1 Find the volume of a sphere with radius 3 m, both in terms of  and to the nearest tenth. Use 3.14 for . = 36 m 3  113.0 m 3 Volume of a sphere Substitute 3 for r. 4343 V = r 3 = (3) 3 4343

97. The surface area of a sphere is four times the area of a great circle.

98. Additional Example 2: Finding Surface Area of a Sphere Find the surface area, both in terms of  and to the nearest tenth. Use 3.14 for . = 36 in 2  113.0 in 2 S = 4r 2 = 4(3 2 ) Surface area of a sphere Substitute 3 for r.

99. Check It Out! Example 2 The moon has a radius of 1738 km. Find the surface area, both in terms of  and to the nearest tenth. Use 3.14 for . = 12,082,576 km 2  37,939,288.6 km 2 S = 4r 2 = 4(1738 2 ) Surface area of a sphere Substitute 1738 for r. 1738 km

100. Additional Example 3: Comparing Volumes and Surface Areas Sphere:  310,464 cm 3 Rectangular Prism: = (44)(84)(84) = 310,464 cm 3 V = lwh V = r 3 = (42 3 ) 4343 4343  74,088 22 7 4343 Compare the volume and surface area of a sphere with radius 42 cm with that of a rectangular prism measuring 44 cm by 84 cm by 84 cm.

101. S = 4r 2 = 4(42 2 ) = 7,056 = 28,896 cm 2 S = 2(44)(84) + 2(44)(84) + 2(84)(84) Additional Example 3 Continued Sphere: Rectangular Prism: S = 2lw + 2lh + 2wh  7,056  22,176 cm 2 22 7 The sphere and the prism have approximately the same volume, but the prism has a larger surface area.

102. Check It Out! Example 3 Sphere:  38,808 mm 3 Rectangular Prism: = (22)(42)(42) = 38,808 mm 3 V = lwh V = r 3 = (21 3 ) 4343 4343  9261 22 7 4343 Compare the volume and surface area of a sphere with radius 21 mm with that of a rectangular prism measuring 22  42  42 mm.

103. S = 4r 2 = 4(21 2 ) = 1764 = 7224 mm 2 S = 2(22)(42) + 2(22)(42) + 2(42)(42) Sphere: Rectangular Prism: S = 2lw + 2lh + 2wh  1764  5544 mm 2 22 7 The sphere and the prism have approximately the same volume, but the prism has a larger surface area. Check It Out! Example 3 Continued

104. Lesson Quiz Find the volume of each sphere, both in terms of  and to the nearest tenth. Use 3.14 for . 1. r = 4 ft2. d = 6 m Find the surface area of each sphere, both in terms of  and to the nearest tenth. Use 3.14 for . 36m 3, 113.0 m 3 85.3ft 3, 267.8 ft 3 2.25 mi 2, 7.1 mi 2 1936 in 2, 6079.0 in 2 3. r = 22 in 4. d = 1.5 mi 5. A basketball has a circumference of 29 in. To the nearest cubic inch, what is its volume? 412 in 3