# Polygons

A is a shape with at least three sides.

## Types of polygon

Polygons can be regular or irregular. If the angles are all equal and all the sides are equal length it is a regular polygon.

## Interior angles of polygons

To find the sum of interior angles in a polygon divide the polygon into triangles.

The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.

### Example

Calculate the sum of interior angles in a pentagon.

A pentagon contains 3 triangles. The sum of the interior angles is:

The number of triangles in each polygon is two less than the number of sides.

The formula for calculating the sum of interior angles is:

(where is the number of sides)

Question

Calculate the sum of interior angles in an octagon.

Using where is the number of sides:

### Calculating the interior angles of regular polygons

All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is:

Question

Calculate the size of the interior angle of a regular .

The sum of interior angles is .

One interior angle is .

## Exterior angles of polygons

If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle.

The sum of the exterior angles of a polygon is 360°.

### Calculating the exterior angles of regular polygons

The formula for calculating the size of an exterior angle is:

Remember the interior and exterior angle add up to 180°.

Question

Calculate the size of the exterior and interior angle in a regular .

Method 1

The sum of exterior angles is 360°.

The exterior angle is .

The interior and exterior angles add up to 180°.

The interior angle is .

Method 2

The sum of interior angles is .

The interior angle is .

The interior and exterior angles add up to 180°.

The exterior angle is .

• The sum of interior angles in a triangle is 180°. To find the sum of interior angles of a polygon, multiply the number of triangles in the polygon by 180°.
• The formula for calculating the sum of interior angles is where is the number of sides.
• All the interior angles in a regular polygon are equal. The formula for calculating the size of an interior angle is: interior angle of a polygon = sum of interior angles ÷ number of sides.
• The sum of exterior angles of a polygon is 360°.
• The formula for calculating the size of an exterior angle is: exterior angle of a polygon = 360 ÷ number of sides.