CK-12 Geometry - Second Edition

http://www.ck12.org Chapter 11. Surface Area and Volume

16

25

=

1536 π

A

16 A= 38400 π

A= 2400 πcm^2

Volumes of Similar Solids

Let’s look at what we know about similar solids so far.

TABLE11.3:

Ratios Units

Scale Factor ab in, ft, cm, m, etc.

Ratio of the Surface Areas

(a

b

) 2

in^2 ,f t^2 ,cm^2 ,m^2 ,etc.

Ratio of the Volumes ?? in^3 ,f t^3 ,cm^3 ,m^3 ,etc.

It looks as though there is a pattern. If the ratio of the volumes follows the pattern from above, it should be the cube
of the scale factor. We will do an example and test our theory.

Example 6:Find the volume of the following rectangular prisms. Then, find the ratio of the volumes.

Solution:

Vsmaller= 3 ( 4 )( 5 ) = 60

Vlarger= 4. 5 ( 6 )( 7. 5 ) = 202. 5

The ratio is 20260. 5 , which reduces to 278 =^2
3
33.
It seems as though our prediction based on the patterns is correct.

Volume Ratio:If two solids are similar with a scale factor ofab, then the volumes are in a ratio of

(a

b

) 3

.

Example 7:Two spheres have radii in a ratio of 3:4. What is the ratio of their volumes?

Solution:If we cube 3 and 4, we will have the ratio of the volumes. Therefore, 3^3 : 4^3 or 27:64 is the ratio of the
volumes.

Example 8:If the ratio of the volumes of two similar prisms is 125:8, what is their scale factor?

Solution:This example is the opposite of the previous example. We need to take the cubed root of 125 and 8 to find
the scale factor.