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.

Maths; Lines and angles; KS3 Year 7; Interior and exterior angles

Example 1

Maths; Lines and angles; KS3 Year 7; Interior and exterior angles

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 $180\degree$ .