# What is the formula for finding exterior and interior angles of a polygon?

Apr 20, 2018

Each interior angle of a regular polygon with $n$ sides:

color(red)(theta = (180(n-2))/n" or "theta = (180n-360)/n

Each exterior angle of a regular polygon with $n$ sides:

color(green)(beta = 180°-theta

Note that interior angle + exterior angle = 180°

theta = 180°-beta " and " beta = 180°-theta

#### Explanation:

To find the size of each interior angle of a regular polygon you need to find the sum of the interior angles first.

If the number of sides is $n$, then

the sum of the interior angles is:

$\textcolor{b l u e}{S = 180 \left(n - 2\right)}$

This formula derives from the fact that if you draw diagonals from one vertex in the polygon, the number of triangles formed will be $2$ less than the number of sides. Each triangle has 180°.

The formula can also be used as $\textcolor{b l u e}{S = 180 n - 360}$

This form of the formula derives from drawing triangles in the polygon by drawing lines from a central point to each vertex. In this way the number of triangles is the same as the number of sides, but the angles at the centre are not required, so 360° is subtracted.

Once you have the sum of all the interior angles you divide by the number of sides to find

the size of each interior angle

$\textcolor{red}{\theta = \frac{180 \left(n - 2\right)}{n}} \text{ or } \textcolor{red}{\theta = \frac{180 n - 360}{n}}$

To find the size of each exterior angle, $\beta$, subtract $\theta$ from 180°

color(green)(beta = 180°-theta

Another method to find the exterior angle is using the fact that the sum of the exterior angles is always 360°

color(green)(beta = (360°)/n

Once you know the size of the exterior angle you can find the size of the interior angle by subtracting from 180°

color(red)(theta = 180°-beta)