10.3. Areas of Similar Polygons http://www.ck12.org
Areas of Similar Polygons
In Chapter 7, we learned about similar polygons. Polygons are similar when the corresponding angles are equal
and the corresponding sides are in the same proportion. In that chapter we also discussed the relationship of the
perimeters of similar polygons. Namely, the scale factor for the sides of two similar polygons is the same as the ratio
of the perimeters.
Example 1:The two rectangles below are similar. Find the scale factor and the ratio of the perimeters.
Solution: The scale factor is^1624 , which reduces to^23. The perimeter of the smaller rectangle is 52 units. The
perimeter of the larger rectangle is 78 units. The ratio of the perimeters is^5278 =^23.
The ratio of the perimeters is the same as the scale factor. In fact, the ratio of any part of two similar shapes
(diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor.
Example 2:Find the area of each rectangle from Example 1. Then, find the ratio of the areas.
Solution:
Asmall= 10 · 16 = 160 units^2
Alarge= 15 · 24 = 360 units^2The ratio of the areas would be^160360 =^49.
The ratio of the sides, or scale factor was^23 and the ratio of the areas is^49. Notice that the ratio of the areas is
thesquareof the scale factor. An easy way to remember this is to think about the units of area, which are always
squared.Therefore, you would alwayssquarethe scale factor to get the ratio of the areas.
Area of Similar Polygons Theorem:If the scale factor of the sides of two similar polygons ismn, then the ratio of
the areas would be
(m
n) 2
.
Example 2:Find the ratio of the areas of the rhombi below. The rhombi are similar.
Solution:There are two ways to approach this problem. One way would be to use the Pythagorean Theorem to find
the length of the 3rdside in the triangle and then apply the area formulas and make a ratio. The second, and easier
way, would be to find the ratio of the sides and then square that.
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5