9.4 Volume of Prisms and Cylinders
9.4 Goal Find the volumes of prisms and cylinders.
Volume of Prisms and Cylinders The amount of water in an aquarium is an example of volume. The volume of a solid is the number of cubic units contained in its interior.
Key Words • prism p. 483 • cylinder p. 485 • volume
EXAMPLE
1
Find the Volume of a Rectangular Prism
Find the volume of the box by determining how many unit cubes fit in the box.
1
Solution The base is 5 units by 3 units. So, 3 p 5, or 15 unit cubes are needed to cover the base layer.
Student Help READING TIP
Volume is measured in cubic units, such as ft 3, read as “cubic feet.”
1
4 units
There are 4 layers. Each layer has 15 cubes. So, the total number of cubes is 4 p 15, or 60. ANSWER
1
unit cube
3 units 5 units
The volume of the box is 60 cubic units.
Volume of a Prism The process used in Example 1 can be used to determine the volume of any prism. Volume of prism
area of base
height h
B
VOLUME OF A PRISM Words
Volume (area of base)(height)
Symbols
500
Chapter 9
Surface Area and Volume
V Bh
h B
Find the Volume of a Prism
2
EXAMPLE
Find the volume of the prism. a.
b.
3 ft
5 in. 4 in.
7 in.
6 ft
8 ft
Solution a. V Bh
Student Help
Write the formula for volume of a prism.
(7 p 4) p 5
Area of rectangular base l p w 7 p 4.
140
Simplify.
ANSWER
The volume is 140 cubic inches.
STUDY TIP b. V Bh
Because you are multiplying three units of measure when you find volume, your answer will always be in cubic units. ft ft ft ft 3
Write the formula for volume of a prism.
12
1 2
p 8 p 6 p 3
Area of triangular base p 8 p 6.
72
Simplify.
ANSWER
The volume is 72 cubic feet.
Find Volume of Prisms Find the volume of the prism. 1.
2.
3.
6 ft 9 ft
7 in.
5 cm
7 in. 10 in.
4 ft
5 cm
5 cm
Volume of a Cylinder The method for finding the volume of
a cylinder is the same for finding the volume of a prism. Volume of cylinder
area of base
height
h
r B πr 2
9.4
Volume of Prisms and Cylinders
501
VOLUME OF A CYLINDER Words
Volume (area of base)(height) h
V Bh πr 2h
Symbols
EXAMPLE
B
r
Compare Volumes of Cylinders
3
2 in.
a. How do the radius and height of
the mug compare to the radius and height of the dog bowl?
4 in. 6 in.
b. How many times greater is the
volume of the bowl than the volume of the mug?
4 in.
Solution a. The radius of the mug is 2 inches and the radius of the dog bowl is
6 inches. The radius of the bowl is three times the radius of the mug. The height of the mug is the same as the height of the bowl.
Volume of dog bowl
b. Volume of mug 2
Write the formula for volume.
π(22)(4)
Substitute for r and for h.
V πr h 16π
V πr 2h π(62)(4) 144π
Simplify.
To compare the volume of the bowl to the volume of the mug, divide the volume of the bowl by the volume of the mug. Volume of bowl 144π 9 Volume of mug 16π ANSWER
The volume of the bowl is nine times the volume of the mug.
Find Volume of Cylinders Find the volume of the cylinder. Round your answer to the nearest whole number. 4.
2 ft
5. 1 in.
6.
4m
5 in. 3 ft
502
Chapter 9
Surface Area and Volume
10 m
9.4 Exercises Guided Practice Vocabulary Check
Based upon the units, tell whether the number is a measure of surface area or volume. 1. 5 ft3
Skill Check
2. 7 yd2
3. 3 m2
4. 2 cm3
Candles Find the volume of the candle. 5.
6.
7.
6 cm 8 cm
12 cm
10 cm
12 cm
B ≈ 63.6 cm2
B ≈ 23.4 cm2
Practice and Applications Extra Practice See p. 692.
Using Unit Cubes Find the number of unit cubes that will fit in the box. Explain your reasoning. 8.
9.
10.
2
3
4
3 5
4 3
4
2
Volume of a Prism Find the volume of the prism. 11.
12.
13.
4 in.
12 m
6 cm 5 in. 5 in.
2 cm
9m
3 cm 4m
Volume of a Cube In Exercises 14–16, you are given the length of each side of a cube. Sketch the cube and find its volume. 14. 3 meters
15. 7 feet
16. 10 centimeters
Visualize It!
In Exercises 17 and 18, make a sketch of the solid. Then find its volume.
Homework Help Example 1: Exs. 8–10 Example 2: Exs. 11–18 Example 3: Exs. 27–40
17. A prism has a square base with 4 meter sides and a height of
7 meters. 18. A prism has a rectangular base that is 3 feet by 6 feet and a height
of 8 feet. 9.4
Volume of Prisms and Cylinders
503
Finding Volume Find the volume of the combined prisms.
Student Help ICLASSZONE.COM
19.
HOMEWORK HELP Extra help with problem solving in Exs. 19–21 is at classzone.com
20.
3 ft
21.
6 in. 2 in.
2m 5m 4m
1 in.
8 ft 4 in.
2 ft
10 m 2 ft
2 in. 1 in.
5 ft
7m
Shopping In Exercises 22–24, use the information about the sizes of the cereal boxes shown below. 22. Find the volume of each box of cereal. 23. How many small boxes
of cereal do you have to buy to equal the amount of cereal in a large box?
Wh
le
$2.00
Cereal
24. Which box gives you the
most cereal for your money? Explain.
Part of a well balanced breakfast
8 in.
ion 1 bow NutritSizecontainer Serving ngper Servi serving t per oun Am s 45 ue Calorie aily Val %D 0g 0g l Fat Fat Tota urated Sat 0mg rol leste g Choium 0mate 2g Sodbohydr Car r 0g Fibe 0g Dietary ars Sug
$6.00
Cereal
ts Fac l
W ho
o le
iner rconta ngpe Servi 45 ries e Calo ily Valu %Da
g l 0m stero Chole er ary Fib Diet rs 0 Suga
0g
16 in.
10 in.
1g than less A *** Protein min Vita m *** Calciu C *** min Vita *** Iron ****** ***** ************ ************ ******
2 in.
Part of a well balanced breakfast
*** ium Calc * ** in C Vitam*** Iron * ******** ****************** ********** ****
10 in.
4 in.
Civil Engineering Soo Locks In Exercises 25 and 26, use the information below.
Lake Superior is about 22 feet higher than Lake Huron. In order for ships to safely pass from one lake to the other, they must go through one of the four Soo Locks. Top View
Lake Huron
SOO LOCKS The first locks
system between Lake Superior and Lake Huron was built around 1797. Today, four locks are available in the Soo Locks system.
Lake Superior
lower gates
Side View
upper gates
80 ft
22 ft Lake Huron
800 ft
Lake Superior
Not drawn to scale 25. Water is added to the MacArthur Lock until the height is increased by
22 feet. To find the amount of water added to the lock, find the volume of a rectangular prism with a length of 800 feet, a width of 80 feet, and a height of 22 feet. 26. How many gallons of water are added to the MacArthur Lock
to raise the ship to the level of Lake Superior? Use the fact that 1 ft3 ≈ 7.5 gal.
504
Chapter 9
Surface Area and Volume
Volume of a Cylinder Find the volume of the cylinder. Round your answer to the nearest whole number. 27.
4 in.
28.
29.
6m
12 m
9 in.
Swimming Pools In Exercises 30–32, find the volume of the pool. Round your answer to the nearest whole number. Then compare the volumes of the pools to answer Exercise 33. 30.
31.
20 ft
32.
24 ft
3 ft
4 ft
4 ft
15 ft
33. Which pool above requires the least amount of water to fill it?
Visualize It!
In Exercises 34 and 35, use the information below. Suppose that a 3-inch by 5-inch index card is rotated around a horizontal line and a vertical line to produce two different solids. 5 in. 3 in. 3 in. 5 in. 34. Find the volume of each solid.
Careers
35. Which solid has a greater volume? Explain your reasoning.
Aquariums In Exercises 36 and 37, use the information below.
The Giant Ocean Tank at the New England Aquarium is a cylinder that is 23 feet deep and 40 feet in diameter as shown.
23 ft
36. Find the volume of the tank. 37. How many gallons of water
are needed to fill the tank? (1 ft3 ≈ 7.5 gal)
AQUARIUM DIVER In
addition to feeding and taking care of the fish and the plants in an aquarium, divers make sure that the tank does not get too crowded.
40 ft
38. Personal Aquariums To avoid overcrowding in a personal
aquarium, you should buy one fish for every gallon of water (231 in.3 ≈ 1 gal). About how many fish should be in an aquarium that is a rectangular prism measuring 20 inches wide, 10 inches long, and is filled with water to a height of 11 inches? 9.4
Volume of Prisms and Cylinders
505
You be the Judge
In Exercises 39 and 40, use the cartons shown. 10 cm
39. Find the volume of each carton of ice cream. 40. Terry assumes that because the
5 cm
BO
10 cm
JUM
Co ol
20 cm Co ol
dimensions doubled, the jumbo carton contains twice as much ice cream as the regular carton. Is Terry right? Explain your reasoning.
Find Volume
EXAMPLE
Find the volume of the passenger car of the Space Spiral at Cedar Point Amusement Park in Sandusky, Ohio.
Solution
14 ft
The passenger car is a cylinder with a “hole” in it. To find the volume, subtract the volume of the hole from the volume of the larger cylinder.
10 ft
4 ft
Volume of larger cylinder πr 2h π(102)(14) ≈ 4398 Volume of “hole” πr 2h π(42)(14) ≈ 704 ANSWER
Student Help
The volume of the passenger car is about 4398 704 3694 cubic feet.
Finding Volume In Exercises 41–43, find the volume of the solid.
ICLASSZONE.COM
41. 2 in.
42.
1 in.
HOMEWORK HELP Extra help with problem solving in Exs. 41–43 is at classzone.com
8 ft
3 ft
43. 1 m
4m
8 in.
10 ft
2 in.
6 in.
4m 4m
Using Algebra Write an expression for the volume of the solid in terms of x. 44.
45.
46.
x
4 x 3 506
Chapter 9
Surface Area and Volume
5x 7
x 2x
3x
Challenge In Exercises 47–49, find the missing dimension(s). If necessary, round your answer to the nearest whole number. 47. A cylinder has a volume of 100.48 cubic inches and a diameter
of 4 inches. Find the height of the cylinder. 48. A cylinder has a volume of 1538.6 cubic feet and a height of
10 feet. Find the radius of the cylinder. 49. The length of a rectangular prism is twice its width. The height
of the prism equals the width. Find the dimensions of the prism, given that the volume is 54 cubic inches.
Standardized Test Practice
50. Multiple Choice What is the approximate volume of the cylinder
shown at the right? A C
3
100 in.
1570 in.3
20 in. B D
3
785 in.
5 in.
6280 in.3
51. Multiple Choice The volume of the prism shown at the right is
168 cubic feet. What is the height of the prism? F H
G J
6 feet 8 feet
7 feet
x
9 feet 3 ft
7 ft
Mixed Review
Using the Pythagorean Theorem Find the unknown side length. Round your answer to the nearest tenth. (Lesson 4.4) 52.
53.
54.
8
12
7
18 b
9 a
14
a
Surface Area Find the surface area of the solid. If necessary, round your answer to the nearest whole number. (Lessons 9.2, 9.3) 55.
3 ft
56.
57.
8m
12 yd
7 ft 9m
5 yd
2m
Algebra Skills
Solving Equations Solve the equation. (Skills Review, p. 672) 58. x 7 0
3 4
61. b 24
59. m 1 12
60. 10 c 3
62. 14d 2
63. 6n 102
9.4
Volume of Prisms and Cylinders
507
Volume of Prisms and Cylinders The amount of water in an aquarium is an example of volume. The volume of a solid is the number of cubic units contained in its interior.
Key Words • prism p. 483 • cylinder p. 485 • volume
EXAMPLE
1
Find the Volume of a Rectangular Prism
Find the volume of the box by determining how many unit cubes fit in the box.
1
Solution The base is 5 units by 3 units. So, 3 p 5, or 15 unit cubes are needed to cover the base layer.
Student Help READING TIP
Volume is measured in cubic units, such as ft 3, read as “cubic feet.”
1
4 units
There are 4 layers. Each layer has 15 cubes. So, the total number of cubes is 4 p 15, or 60. ANSWER
1
unit cube
3 units 5 units
The volume of the box is 60 cubic units.
Volume of a Prism The process used in Example 1 can be used to determine the volume of any prism. Volume of prism
area of base
height h
B
VOLUME OF A PRISM Words
Volume (area of base)(height)
Symbols
500
Chapter 9
Surface Area and Volume
V Bh
h B
Find the Volume of a Prism
2
EXAMPLE
Find the volume of the prism. a.
b.
3 ft
5 in. 4 in.
7 in.
6 ft
8 ft
Solution a. V Bh
Student Help
Write the formula for volume of a prism.
(7 p 4) p 5
Area of rectangular base l p w 7 p 4.
140
Simplify.
ANSWER
The volume is 140 cubic inches.
STUDY TIP b. V Bh
Because you are multiplying three units of measure when you find volume, your answer will always be in cubic units. ft ft ft ft 3
Write the formula for volume of a prism.
12
1 2
p 8 p 6 p 3
Area of triangular base p 8 p 6.
72
Simplify.
ANSWER
The volume is 72 cubic feet.
Find Volume of Prisms Find the volume of the prism. 1.
2.
3.
6 ft 9 ft
7 in.
5 cm
7 in. 10 in.
4 ft
5 cm
5 cm
Volume of a Cylinder The method for finding the volume of
a cylinder is the same for finding the volume of a prism. Volume of cylinder
area of base
height
h
r B πr 2
9.4
Volume of Prisms and Cylinders
501
VOLUME OF A CYLINDER Words
Volume (area of base)(height) h
V Bh πr 2h
Symbols
EXAMPLE
B
r
Compare Volumes of Cylinders
3
2 in.
a. How do the radius and height of
the mug compare to the radius and height of the dog bowl?
4 in. 6 in.
b. How many times greater is the
volume of the bowl than the volume of the mug?
4 in.
Solution a. The radius of the mug is 2 inches and the radius of the dog bowl is
6 inches. The radius of the bowl is three times the radius of the mug. The height of the mug is the same as the height of the bowl.
Volume of dog bowl
b. Volume of mug 2
Write the formula for volume.
π(22)(4)
Substitute for r and for h.
V πr h 16π
V πr 2h π(62)(4) 144π
Simplify.
To compare the volume of the bowl to the volume of the mug, divide the volume of the bowl by the volume of the mug. Volume of bowl 144π 9 Volume of mug 16π ANSWER
The volume of the bowl is nine times the volume of the mug.
Find Volume of Cylinders Find the volume of the cylinder. Round your answer to the nearest whole number. 4.
2 ft
5. 1 in.
6.
4m
5 in. 3 ft
502
Chapter 9
Surface Area and Volume
10 m
9.4 Exercises Guided Practice Vocabulary Check
Based upon the units, tell whether the number is a measure of surface area or volume. 1. 5 ft3
Skill Check
2. 7 yd2
3. 3 m2
4. 2 cm3
Candles Find the volume of the candle. 5.
6.
7.
6 cm 8 cm
12 cm
10 cm
12 cm
B ≈ 63.6 cm2
B ≈ 23.4 cm2
Practice and Applications Extra Practice See p. 692.
Using Unit Cubes Find the number of unit cubes that will fit in the box. Explain your reasoning. 8.
9.
10.
2
3
4
3 5
4 3
4
2
Volume of a Prism Find the volume of the prism. 11.
12.
13.
4 in.
12 m
6 cm 5 in. 5 in.
2 cm
9m
3 cm 4m
Volume of a Cube In Exercises 14–16, you are given the length of each side of a cube. Sketch the cube and find its volume. 14. 3 meters
15. 7 feet
16. 10 centimeters
Visualize It!
In Exercises 17 and 18, make a sketch of the solid. Then find its volume.
Homework Help Example 1: Exs. 8–10 Example 2: Exs. 11–18 Example 3: Exs. 27–40
17. A prism has a square base with 4 meter sides and a height of
7 meters. 18. A prism has a rectangular base that is 3 feet by 6 feet and a height
of 8 feet. 9.4
Volume of Prisms and Cylinders
503
Finding Volume Find the volume of the combined prisms.
Student Help ICLASSZONE.COM
19.
HOMEWORK HELP Extra help with problem solving in Exs. 19–21 is at classzone.com
20.
3 ft
21.
6 in. 2 in.
2m 5m 4m
1 in.
8 ft 4 in.
2 ft
10 m 2 ft
2 in. 1 in.
5 ft
7m
Shopping In Exercises 22–24, use the information about the sizes of the cereal boxes shown below. 22. Find the volume of each box of cereal. 23. How many small boxes
of cereal do you have to buy to equal the amount of cereal in a large box?
Wh
le
$2.00
Cereal
24. Which box gives you the
most cereal for your money? Explain.
Part of a well balanced breakfast
8 in.
ion 1 bow NutritSizecontainer Serving ngper Servi serving t per oun Am s 45 ue Calorie aily Val %D 0g 0g l Fat Fat Tota urated Sat 0mg rol leste g Choium 0mate 2g Sodbohydr Car r 0g Fibe 0g Dietary ars Sug
$6.00
Cereal
ts Fac l
W ho
o le
iner rconta ngpe Servi 45 ries e Calo ily Valu %Da
g l 0m stero Chole er ary Fib Diet rs 0 Suga
0g
16 in.
10 in.
1g than less A *** Protein min Vita m *** Calciu C *** min Vita *** Iron ****** ***** ************ ************ ******
2 in.
Part of a well balanced breakfast
*** ium Calc * ** in C Vitam*** Iron * ******** ****************** ********** ****
10 in.
4 in.
Civil Engineering Soo Locks In Exercises 25 and 26, use the information below.
Lake Superior is about 22 feet higher than Lake Huron. In order for ships to safely pass from one lake to the other, they must go through one of the four Soo Locks. Top View
Lake Huron
SOO LOCKS The first locks
system between Lake Superior and Lake Huron was built around 1797. Today, four locks are available in the Soo Locks system.
Lake Superior
lower gates
Side View
upper gates
80 ft
22 ft Lake Huron
800 ft
Lake Superior
Not drawn to scale 25. Water is added to the MacArthur Lock until the height is increased by
22 feet. To find the amount of water added to the lock, find the volume of a rectangular prism with a length of 800 feet, a width of 80 feet, and a height of 22 feet. 26. How many gallons of water are added to the MacArthur Lock
to raise the ship to the level of Lake Superior? Use the fact that 1 ft3 ≈ 7.5 gal.
504
Chapter 9
Surface Area and Volume
Volume of a Cylinder Find the volume of the cylinder. Round your answer to the nearest whole number. 27.
4 in.
28.
29.
6m
12 m
9 in.
Swimming Pools In Exercises 30–32, find the volume of the pool. Round your answer to the nearest whole number. Then compare the volumes of the pools to answer Exercise 33. 30.
31.
20 ft
32.
24 ft
3 ft
4 ft
4 ft
15 ft
33. Which pool above requires the least amount of water to fill it?
Visualize It!
In Exercises 34 and 35, use the information below. Suppose that a 3-inch by 5-inch index card is rotated around a horizontal line and a vertical line to produce two different solids. 5 in. 3 in. 3 in. 5 in. 34. Find the volume of each solid.
Careers
35. Which solid has a greater volume? Explain your reasoning.
Aquariums In Exercises 36 and 37, use the information below.
The Giant Ocean Tank at the New England Aquarium is a cylinder that is 23 feet deep and 40 feet in diameter as shown.
23 ft
36. Find the volume of the tank. 37. How many gallons of water
are needed to fill the tank? (1 ft3 ≈ 7.5 gal)
AQUARIUM DIVER In
addition to feeding and taking care of the fish and the plants in an aquarium, divers make sure that the tank does not get too crowded.
40 ft
38. Personal Aquariums To avoid overcrowding in a personal
aquarium, you should buy one fish for every gallon of water (231 in.3 ≈ 1 gal). About how many fish should be in an aquarium that is a rectangular prism measuring 20 inches wide, 10 inches long, and is filled with water to a height of 11 inches? 9.4
Volume of Prisms and Cylinders
505
You be the Judge
In Exercises 39 and 40, use the cartons shown. 10 cm
39. Find the volume of each carton of ice cream. 40. Terry assumes that because the
5 cm
BO
10 cm
JUM
Co ol
20 cm Co ol
dimensions doubled, the jumbo carton contains twice as much ice cream as the regular carton. Is Terry right? Explain your reasoning.
Find Volume
EXAMPLE
Find the volume of the passenger car of the Space Spiral at Cedar Point Amusement Park in Sandusky, Ohio.
Solution
14 ft
The passenger car is a cylinder with a “hole” in it. To find the volume, subtract the volume of the hole from the volume of the larger cylinder.
10 ft
4 ft
Volume of larger cylinder πr 2h π(102)(14) ≈ 4398 Volume of “hole” πr 2h π(42)(14) ≈ 704 ANSWER
Student Help
The volume of the passenger car is about 4398 704 3694 cubic feet.
Finding Volume In Exercises 41–43, find the volume of the solid.
ICLASSZONE.COM
41. 2 in.
42.
1 in.
HOMEWORK HELP Extra help with problem solving in Exs. 41–43 is at classzone.com
8 ft
3 ft
43. 1 m
4m
8 in.
10 ft
2 in.
6 in.
4m 4m
Using Algebra Write an expression for the volume of the solid in terms of x. 44.
45.
46.
x
4 x 3 506
Chapter 9
Surface Area and Volume
5x 7
x 2x
3x
Challenge In Exercises 47–49, find the missing dimension(s). If necessary, round your answer to the nearest whole number. 47. A cylinder has a volume of 100.48 cubic inches and a diameter
of 4 inches. Find the height of the cylinder. 48. A cylinder has a volume of 1538.6 cubic feet and a height of
10 feet. Find the radius of the cylinder. 49. The length of a rectangular prism is twice its width. The height
of the prism equals the width. Find the dimensions of the prism, given that the volume is 54 cubic inches.
Standardized Test Practice
50. Multiple Choice What is the approximate volume of the cylinder
shown at the right? A C
3
100 in.
1570 in.3
20 in. B D
3
785 in.
5 in.
6280 in.3
51. Multiple Choice The volume of the prism shown at the right is
168 cubic feet. What is the height of the prism? F H
G J
6 feet 8 feet
7 feet
x
9 feet 3 ft
7 ft
Mixed Review
Using the Pythagorean Theorem Find the unknown side length. Round your answer to the nearest tenth. (Lesson 4.4) 52.
53.
54.
8
12
7
18 b
9 a
14
a
Surface Area Find the surface area of the solid. If necessary, round your answer to the nearest whole number. (Lessons 9.2, 9.3) 55.
3 ft
56.
57.
8m
12 yd
7 ft 9m
5 yd
2m
Algebra Skills
Solving Equations Solve the equation. (Skills Review, p. 672) 58. x 7 0
3 4
61. b 24
59. m 1 12
60. 10 c 3
62. 14d 2
63. 6n 102
9.4
Volume of Prisms and Cylinders
507
