The sum of interior angles of polygons

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

Two irregular polygons.Two irregular polygons.

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

Example

Calculate the sum of the 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 the 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 size of each interior angle of regular polygons

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

\(\text{interior~angle~of~a~regular~polygon}\)\(\text~=~\text{sum~of~interior~angles} \div \text{number~of~sides}\)

Question

Calculate the size of the interior angle of a regular hexagon.

A regular polygon with the equal angles coloured green.

The sum of the interior angles is \(({6}~-~{2})~\times~{180}~=~720^\circ\)

Each interior angle is \({720}~\div~{6}~=~120^\circ\)