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## Similar Polygons: Ratio of Perimeters & Areas

#### High School Geometry - HSG-GPE.B.7

Similar Polygons: Ratio of Perimeters & Areas What if you wanted to create a scale drawing using scale factors? This technique takes a small object, like the handprint below, divides it up into smaller squares, and then blows up the individual squares. Either trace your hand or stamp it on a piece of paper. Then, divide your hand into 9 squares, like the one to the right, probably 2 in × 2 in. Take a larger piece of paper and blow up each square to be 6 in×6 in (meaning you need at least an 18 in a square piece of paper). Once you have your 6 in×6 in squares drawn, use the proportions and area to draw in your enlarged handprint. Polygons are similar when the corresponding angles are equal, and the corresponding sides are in the same proportion. The scale factor for the sides of two similar polygons is the same as the ratio of the perimeters. In fact, the ratio of any part of two similar shapes (diagonals, medians, midsegments, altitudes, etc.) is the same as the scale factor. The ratio of the areas is the square of the scale factor. An easy way to remember this is to think about the units of area, which are always squared. Therefore, you would always square the 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 is mn, then the ratio of the areas would be (m/n)2. These worksheets and lessons help students learn how to take the ratios of similar polygons to find missing or unknown values of those polygons.

### Printable Worksheets And Lessons  #### Homework Sheets

The equation we present often troubles students. Make sure that they clearly understand what each value means.

• Homework 1 - The basic formula that is used is (perimeter of the first triangle divided by the perimeter of the second triangle) is equal to (the largest side of the first triangle divided by the largest side of the second triangle).
• Homework 2 - The areas of two similar polygons are in the ratio 49:25. Find the ratio of the corresponding sides.
• Homework 3 - If two similar triangles have a scale factor of a : b, then the ratio of their areas is a2 : b2.

#### Practice Worksheets

After three years of results, we see that this is the skill that students are having the most trouble with.

• Practice 1 - Two Δ are similar. The sides of the first Δ are 8, 5, and 12. The largest side of the second Δ is 30. Find the perimeter of the second Δ.
• Practice 2 - Find the areas of similar right triangles whose scale factor is 3 : 6.
• Practice 3 - The perimeters of two similar triangles is in the ratio 3 : 5. The sum of their areas is 60 cm2. Find the area of each triangle.

#### Math Skill Quizzes

The quiz questions are presented in the exact form you should see on exams.

• Quiz 1 - The areas of two similar polygons are in the ratio 144:225. Find the ratio of the corresponding sides.
• Quiz 2 - What are the areas of similar right triangles whose scale factor is 2: 6.
• Quiz 3 - Find the ratio of the corresponding sides.