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Surface Area of Similar Solids

# Surface Area of Similar Solids

The figures below are similar.

What is the surface area of the smaller rectangular pyramid?

The following proportion applies to similar solids:

$(\frac{a}{b})^2=\frac{S_1}{S_2}$ where $\frac{a}{b}$ is the ratio of the corresponding dimensions, and $\frac{S_1}{S_2}$ is the ratio of the surface areas.

Find the square of the ratio of the corresponding dimensions:

$(\frac{a}{b})^2=(\frac{3}{7})^2=\frac{9}{49}$

Find the ratio of the surface areas:

$\frac{S_1}{S_2}=\frac{S_1}{196}$

Use these two ratios to set up a proportion and solve for S1.

 $\frac{9}{49}$
=
 $\frac{S_1}{196}$

 9 × 196
=
49
 S1
Find the cross products

1,764 =
49
 S1
Simplify

 1,764 ÷ 49
=
49
 S1
÷ 49
Divide both sides by 49

36 =
 S1

The surface area of the smaller rectangular pyramid is 36 square centimeters. .

The figures below are similar.
What is the surface area of the smaller triangular pyramid?

 2 yd 1 yd 11.32 yd 2

S1 = yd2