## Similar Polygons: Ratio of Perimeters & Areas

#### Aligned To Common Core Standard:

**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

- Sides of Similar Triangles Step-by-Step Lesson- Given two similar triangles, can you find the missing sides of one triangle?
- Guided Lesson - We get more into the concept of ratios when you have similar figures.
- Guided Lesson Explanation - I used very basic ratios here so that they math didn't get in the way of the concept.
- Practice Worksheet - The question jump between polygons and triangles, don't confuse them.
- Matching Worksheet - Teachers always write me about this one. They love it as a warm-up for lesson observations.

#### 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 a
^{2}: b^{2}.

#### 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 cm
^{2}. Find the area of each triangle.

#### Math Skill Quizzes

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