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?

1) 2) 167 min

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.

1) 157.5 yd3

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

2) The volume will be doubled.

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!c

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.

3) 1088∏ in3

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

4) The volume will increased 8 times.

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.

5) 90 cm3

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

1a) 216 cm3

1b) 432√3 m3

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 ft2) 12072 pounds

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)

3a) 360∏ ft3

3b) 45∏ m3

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.

4a) 6 ft3. Volume decreases by 1/64 of the original volume.4b) 756∏ in3 . The volume increases by 27 times.

CONFIDENTIAL 31

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

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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

You did a You did a greatgreat job job today!today!