Perimeter

Perimeter and Area of Polygons

Perimeter and Area of PolygonsThe 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.


scalene triangleExample 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


rectangleExample 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:

perimeter formula where P is the perimeter, L is the length and W is the width.

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


squareExample 3: Find the perimeter of a square with each side measuring 2 inches.

Solution: p.gif = 2 in + 2 in + 2 in + 2 in = 8 in


equilateral triangleExample 4: Find the perimeter of an equilateral triangle with each side measuring 4 centimeters.

Solution: p.gif = 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.


squareExample 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:

p.gif = 4(2 in) = 8 in


equilateral triangleExample 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:

p.gif = 3(4 cm) = 12 cm


pentagonExample 5: Find the perimeter of a regular pentagon with each side measuring 3 inches.

Solution: p.gif = 5(3 in) = 15 in


regular hexagonExample 6: The perimeter of a regular hexagon is 18 centimeters. How long is one side?

Solution: p.gif = 18 cm

Let S 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 (S).

S = 18 cm ÷ 6

S = 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: perimeter_formula. To find the perimeter of a regular polygon, multiply the number of sides by the length of one side.


Exercises

Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. Your answers should be given as whole numbers greater than zero. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR.

1. Find the perimeter of a triangle with sides measuring 10 inches, 14 inches and 15 inches. 
ANSWER BOX:  P =  in  

RESULTS BOX: 

2. A rectangle has a length of 12 centimeters and a width of 4 centimeters. Find the perimeter.
ANSWER BOX:  P =  cm  

RESULTS BOX: 

3. Find the perimeter of a regular hexagon with each side measuring 8 meters.
ANSWER BOX:  P =  m  

RESULTS BOX: 

4. The perimeter of a square is 20 feet. How long is each side?
ANSWER BOX:  s =  ft  

RESULTS BOX: 

5. The perimeter of a regular pentagon is 100 centimeters. How long is each side?
ANSWER BOX:  s =  cm  

RESULTS BOX: