Geometry

Volume of Prisms and Cylinders

04/19/23

Goals

Find the volume of prisms. Find the volume of cylinders. Solve problems using volume.

04/19/23

Volume

The number of cubic units contained in a solid.

Measured in cubic units. Basic Formula:

V = Bh B = area of the base, h = height

04/19/23

Cubic Unit

11

1

V = 1 cu. unit

ss

s

V = s3

04/19/23

Cavalieri’s Principle

04/19/23

B

BB

hh

h

Prism: V = Bh

04/19/23

Cylinder: V = r2h

B

h

r

h

V = Bh

04/19/23

Example 1 Find the volume.

10

8

3

Triangular Prism

V = Bh

Base = 40

V = 40(3) = 120

Abase = ½ (10)(8) = 40

04/19/23

Example 2 Find the volume.

12

10

V = Bh

The base is a ?

Hexagon

04/19/23

Example 2 Solution

12

10

12

?12

?

?

6

12

12 6 3 72

216 3

374.1

A ap

6 3

04/19/23

Example 2 Solution

12

10

374.1

V = Bh

V = (374.1)(10)

V 3741

04/19/23

Example 3A soda can measures 4.5 inches high and the diameter is 2.5 inches. Find the approximate volume.

V = r2h

V = (1.252)(4.5)

V 22 in3

(The diameter is 2.5 in. The radius is 2.5 ÷ 2 inches.)

04/19/23

Example 4A wedding cake has three layers.

The top cake has a diameter of 8 inches, and is 3 inches deep.

The middle cake is 12 inches in diameter, and is 4 inches deep.

The bottom cake is 14 inches in diameter and is 6 inches deep.

Find the volume of the entire cake, ignoring the icing.

04/19/23

Example 4 Solution

8

12

14

3

4

6

r = 4

r = 6

r = 7

VTop = (42)(3) = 48 150.8 in3

VMid = (62)(4) = 144 452.4 in3

VBot = (72)(6) = 294 923.6 in3

486 1526.8 in3

04/19/23

04/19/23

Example 5A manufacturer of concrete sewer pipe makes a pipe segment that has an outside diameter (o.d.) of 48 inches, an inside diameter (i.d.) of 44 inches, and a length of 52 inches. Determine the volume of concrete needed to make one pipe segment.

44

48

52

04/19/23

Example 5 SolutionStrategy:

Find the area of the ring at the top, which is the area of the base, B, and multiply by the height.

View of the Base

04/19/23

Example 5 SolutionStrategy:

Find the area of the ring at the top, which is the area of the base, B, and multiply by the height.

Area of Outer Circle:

Aout = (242) = 576

Area of Inner Circle:

Ain = (222) = 484

Area of Base (Ring):

ABase = 576 - 484 = 92

44

48

52

04/19/23

Example 5 Solution V = Bh

ABase = B = 92

V = (92)(52)

V = 4784

V 15,029.4 in3

44

48

52

04/19/23

Example 5 Alternate Solution

Vouter = (242)(52)

Vouter = 94,096.98

Vinner = (222)(52)

Vinner = 79,067.60

V = Vouter – Vinner

V 15,029.4 in3

44

48

52

04/19/23

Example 6

A metal bar has a volume of 2400 cm3. The sides of the base measure 4 cm by 5 cm. Determine the length of the bar.

4

5L

04/19/23

Example 6 Solution

Method 1 V = Bh B = 4 5 = 20 2400 = 20h h = 120 cm

Method 2 V = L W H 2400 = L 4

5 2400 = 20L L = 120 cm

4

5L

04/19/23

Example 7

3 in

V = 115 in3

A 3-inch tall can has a volume of 115 cubic inches. Find the diameter of the can.

2

2

2

2

2

115 3

115 9.42

115

9.42

12.21

12.21

3.5

V r h

r

r

r

r

r

r

Diameter = 7

04/19/23

Summary

The volumes of prisms and cylinders are essentially the same:

V = Bh & V = r2h where B is the area of the base, h is the

height of the prism or cylinder. Use what you already know about area of

polygons and circles for B.

04/19/23

V = Bh V = r2h

B

h h

r

Add these to your formula sheet.

04/19/23

Which Holds More?

(3.2)(1.6)(4)

20.48

V

3.2 in 1.6 in

4 in 4.5 in

2.3 in

This one!

2

2.34.5

2

18.7

V

04/19/23

What would the height of cylinder 2 have to be to have the same volume as cylinder 1?

r = 4

h

r = 3

8#1#2

04/19/23

Solution

24 8

128

V

r = 4

8#1

04/19/23

Solution

h

r = 3

#2

2128 3

128

914.2

h

h

h

04/19/23

Practice Problems