Perimeter of a Polygon

Key vocabulary

Polygon - a closed 2D shape which is made up of straight lines which are joined together.

Perimeter - The total distance of the outer sides of a closed figure is known as the perimeter. It is the total length of all sides of a polygon. 

What is a polygon?

A polygon is a flat, two-dimensional (2D) shape with straight sides that is fully closed (all the sides are joined up). The sides must be straight. Polygons may have any number of sides.

Shape A is a polygon because it is a 2D shape with straight lines that are fully closed.

Shape B is NOT a polygon because one of the sides is curved

Shape C is NOT a polygon because the shape is not closed. The sides do not join up

Shape A is a special kind of polygon. It is called a regular polygon because all of its lengths and angles are the same.

Shape B is NOT a regular polygon, you can see that the lengths are all different. We call this an irregular polygon.

Perimeter of polygons

A perimeter is the measurement of the distance around a shape

We use length and width measurements to calculate the perimeter of rectangle like the one above

We can simply add up the lengths and the widths to find the perimeter:

Perimeter = 7 + 3 + 7 + 3

Perimeter = 20

We actually have a trick for finding the perimeter of a rectangle. We can use the formula:

Perimeter = 2 x Lengths + 2 x Width

Or

P = 2L + 2W

Let’s use this formula to find out the perimeter of this rectangle:

Let’s start with our formula:

Perimeter = 2 x Lengths + 2 x Width

Or

P = 2L + 2W

For this rectangle:

L = 5cm

W = 2cm

So..

P = 2 x 5 + 2 x 2

We can tidy this up by using brackets

P = (2x5) + (2x2)

Now we can solve this formula, following our BODMAS rule which states that we must start by solving the brackets

P = (10) + (4)

Now that we only have one number inside the brackets we can remove them

P = 10 + 4

P = 14

However, not all polygons are rectangles!

We can categorise polygons into two categories

1 - Regular polygons

2 - Irregular polygons

Regular polygons

A regular polygon is a shape where all the sides have the same length and all the angles are the same. Examples of regular polygons include squares, triangles, and pentagons.

To find the perimeter of a regular polygon, you need to add up the lengths of all the sides.

If all the sides are the same length, you can find the perimeter by multiplying the length of one side by the number of sides.

For example, if you have a square, which has four sides of equal length, and each side is 3 meters long:

you can find the perimeter by multiplying 3 meters by 4, which gives you a perimeter of 12 meters.

The formula for solving the perimeter of a regular polygon is therefore:

Perimeter = number of sides x length of each side

Or

P = N x L

Let's use this formula for another regular polygon:

Here we have a regular polygon in the form of a pentagon.

Let's us the formula

P = N x L to solve the perimeter

N (number of lengths) = 5

L (length of each side) = 5

P = 5 x 5

P = 25

Awesome!

Irregular polygons

Irregular polygons are different! We cannot use a formula because there is no rule for how long a length must be.

The lengths of sides in an irregular polygon can be completely random!

Look at this irregular shape, all the lengths are completely different to each other and there is no pattern between them!

Therefore, the only way this can be done is by measuring each side of the shape, adding up all the lengths of the sides, and then writing down the final answer with the unit of measurement (like inches or centimeters).

So the perimeter of this shape would be:

5 + 6 + 3 + 2 + 2 + 4 = 22cm