Solved Examples: Area Ratios of Similar Polygons with Detailed Steps.

📚 Quick Study Guide

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• If two polygons are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding side lengths.
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• Let $A_1$ and $A_2$ be the areas of two similar polygons, and let $s_1$ and $s_2$ be the lengths of any pair of corresponding sides. Then, $\frac{A_1}{A_2} = \left(\frac{s_1}{s_2}\right)^2$.
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• This also applies to other corresponding linear measures, such as altitudes, medians, and perimeters.
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• To solve area ratio problems, identify corresponding sides or other linear measures, set up the ratio, square it, and use this squared ratio to find the ratio of the areas.

Practice Quiz

1. What is the ratio of the areas of two similar triangles if the ratio of their corresponding sides is 3:5?
   1. 3:5
   2. 9:25
   3. 6:10
   4. 5:3
2. Two similar pentagons have areas of 36 cm² and 81 cm². What is the ratio of their corresponding sides?
   1. 4:9
   2. 2:3
   3. 6:9
   4. 36:81
3. The ratio of the areas of two similar rectangles is 16:25. If the length of the smaller rectangle is 4 cm, what is the length of the larger rectangle?
   1. 5 cm
   2. 6.25 cm
   3. 8 cm
   4. 10 cm
4. Two similar hexagons have side lengths of 5 inches and 10 inches. What is the ratio of their areas?
   1. 1:2
   2. 1:4
   3. 1:8
   4. 1:16
5. A square with side length 6 cm is similar to another square with side length 9 cm. What is the ratio of their areas?
   1. 2:3
   2. 4:9
   3. 3:2
   4. 9:4
6. The area of a smaller similar polygon is 25 cm², and the ratio of the corresponding sides is 2:3. What is the area of the larger polygon?
   1. 37.5 cm²
   2. 56.25 cm²
   3. 75 cm²
   4. 112.5 cm²
7. The perimeters of two similar triangles are 12 cm and 18 cm. What is the ratio of their areas?
   1. 2:3
   2. 4:9
   3. 3:2
   4. 9:4