# How do you find the measure of each exterior angle of a polygon?

Feb 3, 2016

You can find a measure of an exterior angle of a regular polygon with $N$ sides.
It is equal to ${360}^{o} / N$.

#### Explanation:

Angles of a general polygon (exterior and interior) with more than 3 sides are not defined by the lengths of its sides.

However, we can calculate the sum of all interior or exterior angles of any convex polygon. It equals to ${360}^{o}$.

It can be proven geometrically since each exterior angle describes a rotation by some angle and a sum of all exterior angles describes a rotation by full angle of ${360}^{o}$. So, if all exterior angles are equal, like in a regular polygon, each one equals to ${360}^{o} / N$.

It can also be defined with some algebraic calculations based on the fact that a sum of all interior angles is $\left(N - 2\right) \cdot {180}^{o}$.

Dividing the above by $N$ we will obtain a value of an interior angle: $\frac{\left(N - 2\right) \cdot {180}^{o}}{N}$.

Therefore, exterior angle of a regular polygon is
${180}^{o} - \frac{\left(N - 2\right) \cdot {180}^{o}}{N} = {360}^{o} / N$