Geometry

12.4 Volume of Prisms and Cylinders

Geometry 12.4 Volume of Prisms and Cylinders 2Monday, May 5, 2:51

Goals

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

Geometry 12.4 Volume of Prisms and Cylinders 3Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 4Monday, May 5, 2:51

Cubic Unit

11

1

V = 1 cu. unit

ss

s

V = s3

Geometry 12.4 Volume of Prisms and Cylinders 5Monday, May 5, 2:51

Cavalieri’s Principle

Geometry 12.4 Volume of Prisms and Cylinders 6Monday, May 5, 2:51

B

BB

hh

h

Prism: V = Bh

Geometry 12.4 Volume of Prisms and Cylinders 7Monday, May 5, 2:51

Cylinder: V = r2h

B

h

r

h

V = Bh

Geometry 12.4 Volume of Prisms and Cylinders 8Monday, May 5, 2:51

Example 1 Find the volume.

10

8

3

Triangular Prism

V = Bh

Base = 40

V = 40(3) = 120

Abase = ½ (10)(8) = 40

Geometry 12.4 Volume of Prisms and Cylinders 9Monday, May 5, 2:51

Example 2 Find the volume.

12

10

V = Bh

The base is a ?

Hexagon

Geometry 12.4 Volume of Prisms and Cylinders 10Monday, May 5, 2:51

Example 2 Solution

12

10

12

?12

?

?

6

12

12 6 3 72

216 3

374.1

A ap

6 3

Geometry 12.4 Volume of Prisms and Cylinders 11Monday, May 5, 2:51

Example 2 Solution

12

10

374.1

V = Bh

V = (374.1)(10)

V 3741

Geometry 12.4 Volume of Prisms and Cylinders 12Monday, May 5, 2:51

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.)

Geometry 12.4 Volume of Prisms and Cylinders 13Monday, May 5, 2:51

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.

Geometry 12.4 Volume of Prisms and Cylinders 14Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 15Monday, May 5, 2:51

Geometry 12.4 Volume of Prisms and Cylinders 16Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 17Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 18Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 19Monday, May 5, 2:51

Example 5 Solution V = Bh

ABase = B = 92

V = (92)(52)

V = 4784

V 15,029.4 in3

44

48

52

Geometry 12.4 Volume of Prisms and Cylinders 20Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 21Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 22Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 23Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 24Monday, May 5, 2:51

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.

Geometry 12.4 Volume of Prisms and Cylinders 25Monday, May 5, 2:51

V = Bh V = r2h

B

h h

r

Add these to your formula sheet.

Geometry 12.4 Volume of Prisms and Cylinders 26Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 27Monday, May 5, 2:51

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

Geometry 12.4 Volume of Prisms and Cylinders 28Monday, May 5, 2:51

Solution

24 8

128

V

r = 4

8#1

Geometry 12.4 Volume of Prisms and Cylinders 29Monday, May 5, 2:51

Solution

h

r = 3

#2

2128 3

128

914.2

h

h

h