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3. Lesson 13-1 Volumes of Prisms and Cylinders
Lesson 13-2 Volumes of Pyramids and Cones
Lesson 13-3 Volumes of Spheres
Lesson 13-4 Congruent and Similar Solids
Lesson 13-5 Coordinates in Space
Contents

4. Example 1 Volume of a Triangular Prism
Example 2 Volume of a Rectangular Prism
Example 3 Volume of a Cylinder
Example 4 Volume of an Oblique Cylinder
Lesson 1 Contents

5. Find the volume of the triangular prism.
V Bh Volume of a prism
1500 Simplify.
Example 1-1a

6. Find the volume of the triangular prism.
Example 1-1b

7. First, convert feet to inches.
The weight of water is pounds times the volume of water in cubic inches. How many pounds of water would fit into a rectangular child’s pool that is 12 inches deep, 3 feet wide, and 4 feet long?
First, convert feet to inches.
Example 1-2a

8. 20,736 The volume is 20,736 cubic inches.
To find the pounds of water that would fit into the child’s pool, find the volume of the pool.
V Bh Volume of a prism
36(48)(12) B 36(48), h 12
20,736 The volume is 20,736 cubic inches.
Now multiply the volume by
Simplify.
Example 1-2b

9. The weight of water is 62. 4 pounds per cubic foot
The weight of water is 62.4 pounds per cubic foot. How many pounds of water would fit into a back yard pond that is rectangular prism 3 feet deep, 7 feet wide, and 12 feet long?
Example 1-2c

10. Find the volume of the cylinder to the nearest tenth.
The height h is 1.8 centimeters, and the radius r is 1.8 centimeters.
Volume of a cylinder
r 1.8, h 1.8
Use a calculator.
Example 1-3a

11. Find the volume of the cylinder to the nearest tenth.
The diameter of the base, the diagonal, and the lateral edge of the cylinder form a right triangle. Use the Pythagorean Theorem to find the height.
Pythagorean Theorem
a h, b 8, and c 17
Multiply.
Example 1-3b

12. Subtract 64 from each side.
Take the square root of each side.
Now find the volume.
Volume of a cylinder
r 4 and h 15
Use a calculator.
Example 1-3c

13. Find the volume of each cylinder to the nearest tenth. a. b.
Example 1-3d

14. Find the volume of the oblique cylinder to the nearest tenth.
To find the volume, use the formula for a right cylinder.
Volume of a cylinder
r 15, h 25
Use a calculator.
Example 1-4a

15. Find the volume of the oblique cylinder to the nearest tenth.
Example 1-4b

16. End of Lesson 1

17. Example 1 Volume of a Pyramid Example 2 Volumes of Cones
Example 3 Volume of an Oblique Cone
Lesson 2 Contents

18. Teofilo has a solid clock that is in the shape of a square pyramid
Teofilo has a solid clock that is in the shape of a square pyramid. The clock has a base of 3 inches and a height of 7 inches. Find the volume of the clock.
Volume of a pyramid
s 3, h 7 21
Multiply.
Example 2-1a

19. Brad is building a model pyramid for a social studies project
Brad is building a model pyramid for a social studies project. The model is a square pyramid with a base edge of 8 feet and a height of 6.5 feet. Find the volume of the pyramid.
Example 2-1b

20. Find the volume of the cone to the nearest tenth.
Volume of a cone
r = 5, h = 12
Use a calculator.
Example 2-2a

21. Find the volume of the cone to the nearest tenth.
Use trigonometry to find the radius of the base.
Example 2-2b

22. Definition of tangent Solve for r. Use a calculator.
Now find the volume.
Example 2-2c

23. Volume of a cone
Use a calculator.
Example 2-2d

24. Find the volume of each cone to the nearest tenth. a. b.
Example 2-2e

25. Find the volume of the oblique cone to the nearest tenth.
Example 2-3a

26. Volume of a cone
r , h
Use a calculator.
Example 2-3b

27. Find the volume of the oblique cone to the nearest tenth.
Example 2-3c

28. End of Lesson 2

29. Example 1 Volumes of Spheres Example 2 Volume of a Hemisphere
Example 3 Volume Comparison
Lesson 3 Contents

30. Find the volume of the sphere to the nearest tenth.
Volume of a sphere
r = 15
Use a calculator.
Example 3-1a

31. Find the volume of the sphere to the nearest tenth.
First find the radius of the sphere.
Circumference of a circle
C 25
Solve for r.
Example 3-1b

32. Now find the volume.
Volume of a sphere
Use a calculator.
Example 3-1c

33. Find the volume of each sphere to the nearest tenth. a. b.
Example 3-1d

34. Find the volume of the hemisphere.
The volume of a hemisphere is one-half the volume of the sphere.
Volume of a hemisphere
r 3
Use a calculator.
Example 3-2a

35. Find the volume of the hemisphere.
Example 3-2b

36. Short-Response Test Item
Compare the volumes of the sphere and the cylinder with the same radius and height as the radius of the sphere. r
Read the Test Item You are asked to compare the volumes of the sphere and the cylinder.
Example 3-3a

37. Volume of the cylinder: h r
Solve the Test Item
Volume of the sphere:
Volume of the cylinder:
h r
Simplify.
Example 3-3b

38. Short-Response Test Item
Compare the volumes of the hemisphere and the cylinder with the same radius and height as the radius of the hemisphere.
Example 3-3c

39. Example 3-3d

40. End of Lesson 3

41. Example 1 Similar and Congruent Solids
Example 2 Softballs and Baseballs
Lesson 4 Contents

42. Determine whether the pair of solids is similar, congruent, or neither.
Example 4-1a

43. Find the ratios between the corresponding parts of the square pyramids.
Substitution
Simplify.
Example 4-1a

44. Substitution
Simplify.
Substitution
Simplify.
Example 4-1a

45. Determine whether the pair of solids is similar, congruent, or neither.
Example 4-1b

46. Compare the ratios between the corresponding parts of the cones.
Substitution
Simplify.
Substitution
Example 4-1b

47. Determine whether each pair of solids is similar, congruent, or neither.
Example 4-1c

48. b.
Example 4-1d

49. Write the ratio of the corresponding measures of the balls.
Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches. Find the scale factor of the two balls.
Write the ratio of the corresponding measures of the balls.
Substitution
Simplify.
Example 4-2a

50. If the scale factor is a:b, then the ratio of the surface areas is
Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches. Find the ratio of the surface areas of the two balls.
If the scale factor is a:b, then the ratio of the surface areas is
Example 4-2b

51. Theorem 13.1
Simplify.
Example 4-2b

52. If the scale factor is a : b, then the ratio of the volumes is
Softballs and baseballs are both used to play a game with a bat. A softball has a diameter or 3.8 inches, while a baseball has a diameter of about 2.9 inches Find the ratio of the volumes of the two balls.
If the scale factor is a : b, then the ratio of the volumes is
Example 4-2c

53. Theorem 13.1
Simplify.
Example 4-2c

54. Two sizes of balloons are being used for decorating at a party
Two sizes of balloons are being used for decorating at a party. When fully inflated, the balloons are spheres. The first balloon has a diameter of 18 inches while the second balloon has a radius of 7 inches.
Example 4-2d

55. a. Find the scale factor of the two balloons.
b. Find the ratio of the surface areas of the two balloons.
c. Find the ratio of the volumes of the two balloons.
Example 4-2e

56. End of Lesson 4

57. Example 1 Graph of a Rectangular Solid
Example 2 Distance and Midpoint Formulas in Space
Example 3 Translating a Solid
Example 4 Dilation with Matrices
Lesson 5 Contents

58. Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex.
Plot the x-coordinate first. Draw a segment from the origin 3 units in the negative direction.
Example 5-1a

59. Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex.
To plot the y-coordinate, draw a segment 1 unit in the positive direction.
Example 5-1a

60. Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex.
Next, to plot the z-coordinate, draw a segment 2 units long in the positive direction.
Example 5-1a

61. Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex.
Label the coordinate A.
Example 5-1a

62. Draw the rectangular prism and label each vertex.
Graph the rectangular solid that contains the ordered triple A(–3, 1, 2) and the origin. Label the coordinates of each vertex.
Answer:
Draw the rectangular prism and label each vertex.
Example 5-1a

63. Graph the rectangular solid that contains the ordered triple N (1, 2, –3) and the origin. Label the coordinates of each vertex.
Answer:
Example 5-1b

64. Determine the distance between F(4, 0, 0) and G(–2, 3, –1).
Distance Formula in Space
Substitution
Simplify.
Answer:
Example 5-2a

65. Determine the midpoint M of
Midpoint Formula in Space
Substitution
Simplify.
Example 5-2b

66. a. Determine the distance between A(0, –5, 0) and B(1, –2, –3).
b. Determine the midpoint M of
Answer:
Example 5-2b

67. Suppose a two-story home is built with a bathroom on the first floor that is 9 feet wide, 6 feet deep, and 8 feet tall. Likewise, a bathroom on the second floor is directly above the one on the first floor and has the same dimensions. If the second floor is 10 feet above the first floor, find the coordinates of each vertex of the rectangular prism that represents the second floor bathroom.
Explore Since the bathroom is a rectangular prism, use positive values for x, y, and z. Write the coordinates of each corner. The points of the bathroom will rise 10 feet for the points of the second-floor bathroom.
Example 5-3a

68. Plan Use a translation equation (x, y, z) (x, y, z + 10) to find the coordinates of each vertex of the rectangular prism that represents the second-floor bathroom.
Solve
(6, 0, 10)
D(6, 0, 0)
(6, 9, 10)
C(6, 9, 0)
(0, 9, 10)
B(0, 9, 0)
(0, 0, 10)
A(0, 0, 0)
Translated coordinates, (x, y, z + 10) Image
Coordinates of the vertices, (x, y, z) Preimage
(0, 0, 18)
E(0, 0, 8)
(0, 9, 18)
F(0, 9, 8)
(6, 9, 18)
G(6, 9, 8)
(6, 0, 18)
H(6, 0, 8)
Example 5-3a

69. Examine Check that the distance between corresponding vertices is 10 feet.
Answer:
Example 5-3a

70. Suppose a warehouse has a room on the ground floor that is 20 feet wide, 25 feet long, and 12 feet tall. If the height of each floor is 12 feet, find the coordinates of each vertex of the rectangular prism that represents a room in the basement of the warehouse directly below the given room.
Example 5-3b

71. Dilate the prism by a scale factor of . Graph the image under
the dilation.
First, write a vertex matrix for the rectangular prism. 2 4 H G F E D C B A z y x
Example 5-4a

72. Next, multiply each element of the vertex matrix by the
scale factor, . = 1 2 4 H G F E D C B A
Example 5-4a

73. Answer:
The coordinates of the vertices are A'(0, 0, 0), B'(0, 2, 0), C'(1, 2, 0), D'(1, 0, 0), E'(1, 0, 1), F'(0, 0, 1), G'(0, 2, 1), H'(1, 2, 1).
Example 5-4a

74. Dilate the prism by a scale factor of 3
Dilate the prism by a scale factor of 3. Graph the image under the dilation.
Example 5-4b

75. Answer:
The coordinates of the vertices are A'(0, 0, 0), B'(0, 3, 3), C'(0, 3, 0), D'(3, 3, 3), E'(0, 0, 3), F'(3, 0, 0), G'(3, 3, 0), H'(3, 0, 3).
Example 5-4b

76. End of Lesson 5