# Finding Perimeter & Area of Similar Polygons

Instructor: David Karsner

David holds a Master of Arts in Education

Learn how to find the perimeter and area of similar polygons by using the scale factor between the two similar polygons. Work examples for both perimeter and area.

## Decorating Two Similar Polygon Rooms

Suppose you are decorating two rooms that are the same shape except one is a little larger than the other. You are going to add a border along the ceilings and carpet both rooms. Since the dimensions of the larger room are 20% larger than the smaller room, you assume that you will need 20% more border and carpet for the larger room. Is this an accurate assumption? Well, let's learn how to find the perimeter and area of similar polygons so we can figure that out.

## Vocabulary

A polygon is a two-dimensional shape that is closed, meaning the sides completely enclose an area, and it has line segments for sides. Triangles, rectangles, and trapezoids are all polygons. A circle is not a polygon because it doesn't have line segments for sides.

Similar polygons are polygons that have corresponding angles, (a pair of angles from each polygon in the same position), that are congruent, (have the same measurement), and the corresponding sides are proportional. Proportional means they have a common ratio. This common ratio is also called the scale factor. In summary, polygons are similar when they have the exact same shape and their interior angles are the same and their sides are proportional.

Perimeter is the one-dimensional measurement of the distance around a shape. You can find the perimeter of any polygon by adding the length of all the sides.

Area is the two-dimensional measurement of the amount of space inside a shape. It is always measured in square units. Finding the area of a polygon typically involves multiplying the length and width together in some manner.

## Perimeter of Similar Polygons

When two polygons are similar, the sides have a common ratio. If you multiply the length of one side of the polygon by the scale factor (let's call it r), you will get the length of the corresponding side of the other polygon. This process is applied to all the sides of the polygon.

Let's say we have a triangle with sides of 3, 4, and 7 units and a perimeter of 14 units. We have another triangle that is similar and has a scale factor of r to the first triangle. The corresponding sides of the second triangle are 3r, 4r, and 7r units with a perimeter of 14r units. Notice that the only difference between the perimeter of the first triangle and the perimeter of the second triangle is multiplication by r.

### Example 1

Triangles ABC and DEF are similar triangles. What is the perimeter of triangle DEF? To determine this, we need to find the scale factor between sides of the triangles. Since we have the measurements of two corresponding sides, we can easily do that. DE and AB are corresponding sides. The length of DE is 12 and the length of AB is 4, so their ratio and scale factor is 12/4 or 3. Since each side of triangle DEF is 3 times each side of triangle ABC, the perimeter of triangle DEF is 3 times greater than the perimeter of triangle ABC.

Because we are given the measurements of each side of triangle ABC, we can find that the perimeter of triangle ABC is 11 (4 + 4 + 3 = 11). Now we can multiply that perimeter by the scale factor of 3 to determine the perimeter of triangle DEF (11 x 3 = 33).

## Area of Similar Polygons

Using the scale factor is different between perimeter and area because perimeter is one dimensional and area is two dimensional. Let's say we have two similar rectangles with a scale factor of r. The first rectangle has dimensions of 5 by 6 units, with an area of 30 square units. The dimension of the second rectangle would by 5r by 6r units with an area of (30) x (r) x (r) or 30r2 square units. Notice how the scale factor (r) is squared to find the area of the similar rectangle.

### Example 2

Rectangles ABCD and EFGH are similar rectangles. What is the area of rectangle ABCD?

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