Lesson on Perimeter of Polygons

The perimeter of a polygon is the distance around the outside of the polygon. A polygon is 2-dimensional; however, perimeter is 1-dimensional and is measured in linear units. To help us make this distinction, look at our picture of a rectangular backyard. The yard is 2-dimensional: it has a length and a width. The amount of fence needed to enclose the backyard (perimeter) is 1-dimensional. The perimeter of this yard is the distance around the outside of the yard, indicated by the red arrow; It is measured in linear units such as feet or meters.

To find the perimeter of a polygon, take the sum of the length of each side. The polygons below are much smaller than a fenced-in yard. Thus, we use smaller units in our examples, such as centimeters and inches.

Example 1:	Find the perimeter of a triangle with sides measuring 5 centimeters, 9 centimeters and 11 centimeters.
Solution:	P = 5 cm + 9 cm + 11 cm = 25 cm

Example 2:	A rectangle has a length of 8 centimeters and a width of 3 centimeters. Find the perimeter.
Solution 1:	P = 8 cm + 8cm + 3 cm + 3 cm = 22 cm
Solution 2:	P = 2(8 cm) + 2(3 cm) = 16 cm + 6 cm = 22 cm

In Example 2, the second solution is more commonly used. In fact, in mathematics, we commonly use the following formula for perimeter of a rectangle:

,  where is the perimeter, is the length and is the width.

In the next few examples, we will find the perimeter of other polygons.

Example 3:	Find the perimeter of a square with each side measuring 2 inches.
Solution:	= 2 in + 2 in + 2 in + 2 in = 8 in

Example 4:	Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.
Solution:	= 4 cm + 4 cm + 4 cm = 12 cm

A square and an equilateral triangle are both examples of regular polygons. Another method for finding the perimeter of a regular polygon is to multiply the number of sides by the length of one side. Let's revisit Examples 3 and 4 using this second method.

Example 3:	Find the perimeter of a square with each side measuring 2 inches.
Solution:	This regular polygon has 4 sides, each with a length of 2 inches. Thus we get:
	= 4(2 in) = 8 in

Example 4:	Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.
Solution:	This regular polygon has 3 sides, each with a length of 4 centimeters. Thus we get:
	= 3(4 cm) = 12 cm
Example 5:	Find the perimeter of a regular pentagon with each side measuring 3 inches.
Solution:	= 5(3 in) = 15 in
Example 6:	The perimeter of a regular hexagon is 18 centimeters. How long is one side?
Solution:	= 18 cm
	Let represent the length of one side. A regular hexagon has 6 sides, so we can divide the perimeter by 6 to get the length of one side ().
	= 18 cm ÷ 6
	= 3 cm
Summary:	To find the perimeter of a polygon, take the sum of the length of each side. The formula for perimeter of a rectangle is: . To find the perimeter of a regular polygon, multiply the number of sides by the length of one side.