# 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:

$180 \times 3 = 540^\circ$

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:

$$(n - 2) \times 180^\circ$$ (where $$n$$ is the number of sides)

Question

Calculate the sum of interior angles in an octagon.

Using $$(n - 2) \times 180^\circ$$ where $$n$$ is the number of sides:

$(8 - 2) \times 180 = 1,080^\circ$

### 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:

$\text{interior angle of a polygon} = \text{sum of interior angles} \div \text{number of sides}$

Question

Calculate the size of the interior angle of a regular .

The sum of interior angles is $$(6 - 2) \times 180 = 720^\circ$$.

One interior angle is $$720 \div 6 = 120^\circ$$.

## 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:

$\text{exterior angle of a polygon} = 360 \div \text{number of sides}$

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 $$360 \div 5 = 72^\circ$$.

The interior and exterior angles add up to 180°.

The interior angle is $$180 - 72 = 108^\circ$$.

Method 2

The sum of interior angles is $$(5 - 2) \times 180 = 540^\circ$$.

The interior angle is $$540 \div 5 = 108^\circ$$.

The interior and exterior angles add up to 180°.

The exterior angle is $$180 - 108 = 72^\circ$$.

• 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 $$(n - 2) \times 180^\circ$$ where $$n$$ 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.