Similarity: Similar Polygons

Perimeter and Area of Similar Polygons

Let’s look at two similar polygons and compare their perimeters and areas.  We will start with basic squares and then find a general rule for all similar polygons.

Take a look at the similar squares below.

First square with side lengths 4; second square with side length 6.
Two squares

 

The ratio of the side lengths of the first square to the second square is 4:6 or four-sixths equals two-thirds.

The perimeter of each is:
P1 = 4 + 4 + 4 + 4 = 16
P2 = 6 + 6 + 6 + 6 = 24

The ratio of the perimeters is 16:24 or sixteen-twenty-fourths equals two-thirds.

Notice that the ratio of the perimeters is the same as the ratio of the sides.

 

The area of each is:
A1 = 4(4) = 16
A2 = 6(6) = 36

The ratio of the areas is 16:36 or sixteen-thirty-sixths equals four-ninths.

Notice that the ratio of the areas is the square of the ratio of the sides.

We can generalize this to all polygons.

If the ratio of the sides of two similar polygons is A to B, the ratio of the perimeters is also A to B and the ratio of the areas is A² to B².

 

Ratio of sides: the fraction A over B
Ratio of perimeters: the fraction A over B
Ratio of areas: the fraction A squared over B squared