CK-12 Geometry - Second Edition

11.7. Exploring Similar Solids http://www.ck12.org

Solution:

SAsmaller= 2 ( 4 · 3 )+ 2 ( 4 · 5 )+ 2 ( 3 · 5 )

= 24 + 40 + 30 = 94 units^2

SAlarger= 2 ( 6 · 4. 5 )+ 2 ( 4. 5 · 7. 5 )+ 2 ( 6 · 7. 5 )

= 54 + 67. 5 + 90 = 211. 5 units^2

Now, find the ratio of the areas. 21194. 5 =^49 =^2
2
32. The sides are in a ratio of

4

6 =

2
3 , so the surface areas have the same
relationship as the areas of two similar shapes.

Surface Area Ratio:If two solids are similar with a scale factor ofab, then the surface areas are in a ratio of

(a

b

) 2

.

Example 4: Two similar cylinders are below. If the ratio of the areas is 16:25, what is the height of the taller
cylinder?

Solution:First, we need to take the square root of the area ratio to find the scale factor,

√

16

25

=^45. Now we can set

up a proportion to findh.

4

5

=

24

h

4 h= 120

h= 30

Example 5:Using the cylinders from Example 4, if the surface area of the smaller cylinder is 1536πcm^2 , what is
the surface area of the larger cylinder?

Solution:Set up a proportion using the ratio of the areas, 16:25.