Interior and exterior angles

​​In a nutshell

Interior and exterior angles refer to specific angles in a shape (or polygon). There are formulae that can be used to calculate the interior and exterior angles of regular polygons.

Definitions

A polygon is another word to describe a 2D shape.

A regular polygon is a 2D shape that has equal sides and angles.

An interior angle of a polygon is the inner angle formed by two sides of a shape.

An exterior angle of a polygon is the formed by extending one side of the shape past the vertex.

Example 1

In the diagram above, the angles $$\alpha$$ and $$75\degree$$ are internal angles. The angle $$135^\circ$$ is an external angle.

Note: It is worth noticing that an interior and exterior angle are two angles on a straight line, which means that they add together to make $$180\degree$$.

Finding the interior angles of a regular polygon

The sum of the interior angles of a polygon can be found using the formula:

​$$\theta=(n-2)\times180$$​​

Where $$\theta$$​ is the sum of the interior angles, and $$n$$​ is the number of sides.

The interior angles of a regular polygon are equal. Therefore, to find the size of one interior angle, divide the value of $$\theta$$ by the number of sides.

Example 2

What is the size of an interior angle of a regular nonagon?

A nonagon has $$9$$ sides, so $$n=9$$. The sum of the interior angles is therefore given to be:

$$\theta=(n-2)\times180=(9-2)\times180=7\times180$$​​

$$\theta=1260\degree$$​​

To find the size of one interior angle, divide this number by $$9$$:

$$1260\div9=140$$​​

The size of one interior angle of a nonagon is $$\underline{ 140\degree}$$.

​

Finding the exterior angles of a regular polygon

Exterior angles of a regular polygon always add up to $$360\degree$$. Therefore, the formula for one exterior angle is given to be:

​$$\theta=\dfrac{360}{n}$$​​

Where $$\theta$$​ is the size of one exterior angle, and $$n$$​ is the number of sides.

Example 3

What is the size of one exterior angle of a regular nonagon?

Substitute $$n=9$$ into the formula to give:

​$$\theta=\dfrac{360}{9}=40$$​​

The size of one exterior angle of a regular nonagon is $$\underline{ 40\degree}$$.

Note: Notice how the interior and exterior angle of a regular nonagon adds up to $$180\degree$$.