CONFIDENTIAL 1
GeometryGeometry
Volume of Prisms Volume of Prisms and Cylinders and Cylinders
CONFIDENTIAL 2
Warm UpWarm Up
1) Marcy, Rachel, and Tina went bowling. Marcy bowled 100 less than twice Rachel’s score. Tina bowled 40 more than Rachel’s score. Rachel bowled a higher score than Marcy. What is the greatest score that Tina could have bowled?
2) Max can type 40 words per minute. He estimates that his term paper contains about 5000 words, and he takes a 15-minute break for every 45 minutes of typing. About how much time will it take Max to type his term paper?
CONFIDENTIAL 3
Volume of Prisms and Cylinders
A cube built out of 27 unit cubes has a volume of 27 cubic units.
The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill
the interior.
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CONFIDENTIAL 4
A right prism and an oblique prism with the same base and height have the same volume
Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-
sectional area at every level, they have the same volume.
CONFIDENTIAL 5
Volume of a Prism
h
B B
h
The volume of a prism with base area B and height h is V = Bh.
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CONFIDENTIAL 6
s
s
s
h
w
l
The volume of a right rectangular prism with length l, width w, and height h is V = lwh.
The volume of a cube with edge length s is V = s . 3
CONFIDENTIAL 7
Finding Volumes of Prisms
Find the volume of each prism. Round to the nearest tenth, if necessary.
A).
10 cm
12 cm
8 cm
volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh.
V = lwh = (10)(12)(8) = 980 cm
3
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CONFIDENTIAL 8
10 cm
12 cm
8 cm
B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s.33
3
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CONFIDENTIAL 9
C). A right regular pentagonal Prism with base edge length 5 m and height 7 m.
36°
7 m
5 m
Step:1Step:1 Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36°
tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m.a = 2.5/tan 36° Solve for a.
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CONFIDENTIAL 10
36°
7 m
5 m
Step:2Step:2
Step:3Step:3
Use the value of a to find the base area.
Use the base area to find the volume.
B = 1
2
2.5
tan 36 25 =
31.25
tan 36 P = 5(5) = 25 m
V = Bh = 31.25
tan 36 7 301.1 m3
CONFIDENTIAL 11
Now you try!
1) Find the volume of a triangular prism with a height of 9 yd whose base is a right triangle with
legs 7 yd and 5 yd long.
CONFIDENTIAL 12
Marine Biology Application
The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in
gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.
(Hint: 1 gallon = 0.134 ft )3
8 ft
120 ft
60 ft
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CONFIDENTIAL 13
8 ft
120 ft
60 ft
Step:1Step:1 Find the volume of the aquarium in cubic feet.
V = lwh = (120)(60)(8) = 57,600 cm 3
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CONFIDENTIAL 14
8 ft
120 ft
60 ft
Step:2Step:2 Use the conversion factor 1 gallon
0.134 ft3 to estimate the
volume in gallons.
57,600 ft3 1 gallon
0.134 ft3 = 429,851 gallons
1 gallon
0.134 ft3 = 1
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CONFIDENTIAL 15
Step:3Step:3
Use the conversion factor 8.33 pounds
1 gallon to estimate the weight of the
water.
429,851 gallons 8.33 pounds
1 gallon 3,580,659 pounds
8.33 pounds
1 gallons = 1
The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds
8 ft
120 ft
60 ft
CONFIDENTIAL 16
Now you try!
2) Estimate the volume in gallons and the weight of the water in the aquarium below if the height were doubled.
8 ft
120 ft
60 ft
CONFIDENTIAL 17
Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they
have the same volume.
CONFIDENTIAL 18
r
h
r
h
The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h.2
CONFIDENTIAL 19
12 cm
8 cm
Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth.
A).
V = r2h Volume of a cylinder = 8 2 12 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3
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Finding Volumes of Cylinders
CONFIDENTIAL 20
12 cm
8 cm
B). A cylinder with a base area of 36 in and a height equal to twice the radius.
2
Step:1Step:1 Use the base area to find the radius.
Step:3Step:3
Step:2Step:2 Use the radius to find the height. The height is equal to twice the radius.
Use the radius and height to find the volume.
r2 = 36 Substitute 36 for the base area. r = 6 Solve for r.
h = 2r = 2(6) = 12cm
V = r2h Volume of a cylinder = (6)2(12) = 432 in3 Substitute 6 for r and 12 for h. 1357.2 in3
CONFIDENTIAL 21
Now you try!
3). Find the volume of a cylinder with a diameter of 16in. and a height of 17 in. Give your answer both in terms of and
rounded to the nearest tenth.
CONFIDENTIAL 22
Exploring Effects of Changing Dimensions
Exploring Effects of Changing Dimensions
The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume.
6 m
12 moriginal dimensions: radius and height multiplied by1
2.
V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3
Notice that 54 = 1
8(432). if the radius and height are multiplied by
1
2, the volume is multiplied by
1
2 3
, or 1
8.
CONFIDENTIAL 23
Now you try!
4) The length, width, and height of the prism are doubled. Describe the effect on the volume.
1.5 ft
4 ft 3 ft
CONFIDENTIAL 24
Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
5 m
8 m
9 m
6 m
The base area of the prism is
B = 1
2(6)(8) =24 m2.
The volume of the prism is V = Bh = 24(9) = 216 m3.The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m.The volume of the cylinder is V = r2h = (5)2(5) = 125m3.The total volume of the figure is the sum of the volumes.V = 216 + 125 608.7 m3
CONFIDENTIAL 25
Now you try!
3 cm
5 cm
5) Find the volume of the composite figure. Round to the nearest tenth.
CONFIDENTIAL 26
Now some problems for you to practice !
CONFIDENTIAL 27
Assessment
1. Find the volume of each prism.
A) B)
6 cm
4 cm
9 cm
6 m
8 m
CONFIDENTIAL 28
2. The world’s largest ice cream cake, built in New York City on may 25, 2004, was approximately a 19 ft by 9 ft by 2 ft rectangular prism. Estimate the volume of the ice cream
cake in gallons. If the density of the ice cream cake was 4.73 pounds per gallon, estimate the weight of the cake.
(Hint: 1 gallon = 0.134 cubic feet)
2 ft
19 ft
9 ft
CONFIDENTIAL 29
3. Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth.
12 ft
10 ft 5 m
3 m
A) B)
CONFIDENTIAL 30
4. Describe the effect of each change on the volume of the given figure.
7 in.
2 in.
8 ft
4 ft
12 ft
A) The dimensions are multiplied by ¼ .
B) The dimensions are tripled.
CONFIDENTIAL 31
5. Find the volume of each composite figure. Round to the nearest tenth.
A) B)
14 ft
6 ft
4 ft
4 ft
12 ft
15 in.
10 in. 5 in.
CONFIDENTIAL 32
Let’s review
CONFIDENTIAL 33
Volume of Prisms and Cylinders
A cube built out of 27 unit cubes has a volume of 27 cubic units.
The volume of a three-dimensional figure is the number of nonoverlapping unit cubes of a given size that will exactly fill
the interior.
Next Page:
CONFIDENTIAL 34
A right prism and an oblique prism with the same base and height have the same volume
Cavalieri's principle says that if two three-dimensional figure have the same height and have the same cross-
sectional area at every level, they have the same volume.
CONFIDENTIAL 35
Volume of a Prism
h
B B
h
The volume of a prism with base area B and height h is V = Bh.
Next Page:
CONFIDENTIAL 36
s
s
s
h
w
l
The volume of a right rectangular prism with length l, width w, and height h is V = lwh.
The volume of a cube with edge length s is V = s . 3
CONFIDENTIAL 37
Finding Volumes of Prisms
Find the volume of each prism. Round to the nearest tenth, if necessary.
A).
10 cm
12 cm
8 cm
volume of a right rectangular prism Substitute 10 for l, 12 for w, and 8 for h is V = lwh.
V = lwh = (10)(12)(8) = 980 cm
3
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CONFIDENTIAL 38
10 cm
12 cm
8 cm
B). A cube with edge length 10 cm V= s Volume of a cube = 10 = 1000 cm Substitute 10 for s.33
3
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CONFIDENTIAL 39
C). A right regular pentagonal Prism with base edge length 5 m and height 7 m.
36°
7 m
5 m
Step:1Step:1 Find the apothem a of the base . First draw a right triangle on one base as shown. The measure of the angle with its vertex at the center is 360°/10 =36°
tan 36° = 2.5/a The leg of the triangle is half the side length. Or 2.5 m.a = 2.5/tan 36° Solve for a.
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CONFIDENTIAL 40
36°
7 m
5 m
Step:2Step:2
Step:3Step:3
Use the value of a to find the base area.
Use the base area to find the volume.
B = 1
2
2.5
tan 36 25 =
31.25
tan 36 P = 5(5) = 25 m
V = Bh = 31.25
tan 36 7 301.1 m3
CONFIDENTIAL 41
Marine Biology Application
The aquarium at the right is a rectangular prism. Estimate the volume of the water in the aquarium in gallons. The density of water is about 8.33 pounds per gallon. Estimate the weight of the water in pounds.
(Hint: 1 gallon = 0.134 ft )3
8 ft
120 ft
60 ft
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CONFIDENTIAL 42
8 ft
120 ft
60 ft
Step:1Step:1 Find the volume of the aquarium in cubic feet.
V = lwh = (120)(60)(8) = 57,600 cm 3
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CONFIDENTIAL 43
8 ft
120 ft
60 ft
Step:2Step:2 Use the conversion factor 1 gallon
0.134 ft3 to estimate the
volume in gallons.
57,600 ft3 1 gallon
0.134 ft3 = 429,851 gallons
1 gallon
0.134 ft3 = 1
Next Page:
CONFIDENTIAL 44
Step:3Step:3
Use the conversion factor 8.33 pounds
1 gallon to estimate the weight of the
water.
429,851 gallons 8.33 pounds
1 gallon 3,580,659 pounds
8.33 pounds
1 gallons = 1
The aquarium holds about 429,851 gallons. The water in the aquarium weight about 3,580,659 pounds
8 ft
120 ft
60 ft
CONFIDENTIAL 45
Cavalieri’s principle also relates to cylinders. The two stacks have the same number of CDs, so they
have the same volume.
CONFIDENTIAL 46
r
h
r
h
The volume of a cylinder with base area B, radius r, and height h is V = Bh, or V = r h.2
CONFIDENTIAL 47
12 cm
8 cm
Find the volume of each cylinder. Give your answers both in terms of and rounded to the nearest tenth.
A).
V = r2h Volume of a cylinder = 8 2 12 Substitute 8 for r and 12 for h. = 768 cm3 2412.7 cm3
Next Page:
Finding Volumes of Cylinders
CONFIDENTIAL 48
12 cm
8 cm
B). A cylinder with a base area of 36 in and a height equal to twice the radius.
2
Step:1Step:1 Use the base area to find the radius.
Step:3Step:3
Step:2Step:2 Use the radius to find the height. The height is equal to twice the radius.
Use the radius and height to find the volume.
r2 = 36 Substitute 36 for the base area. r = 6 Solve for r.
h = 2r = 2(6) = 12cm
V = r2h Volume of a cylinder = (6)2(12) = 432 in3 Substitute 6 for r and 12 for h. 1357.2 in3
CONFIDENTIAL 49
Exploring Effects of Changing Dimensions
Exploring Effects of Changing Dimensions
The radius and height of the cylinder are multiplied by ½. Describe the effect on the volume.
6 m
12 moriginal dimensions: radius and height multiplied by1
2.
V = r2h V = r2h = (6)2(12) = (3)2(6) = 432 m3 =54 m3
Notice that 54 = 1
8(432). if the radius and height are multiplied by
1
2, the volume is multiplied by
1
2 3
, or 1
8.
CONFIDENTIAL 50
Finding Volumes of Composite Three-Dimensional Figures
Find the volume of the composite figure. Round to the nearest tenth.
5 m
8 m
9 m
6 m
The base area of the prism is
B = 1
2(6)(8) =24 m2.
The volume of the prism is V = Bh = 24(9) = 216 m3.The cylinder's diameter equals the hypotenuse of the prism's base, 10 m. So the radius is 5 m.The volume of the cylinder is V = r2h = (5)2(5) = 125m3.The total volume of the figure is the sum of the volumes.V = 216 + 125 608.7 m3
CONFIDENTIAL 51
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