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Interior and exterior angles

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# 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 $180\degree$.

## FAQs - Frequently Asked Questions

### What are interior and exterior angles?

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