# What is the sum of the exterior angles of a polygon with 4 sides? 5 sides? 6 sides? n sides?

Jul 29, 2018

The sum of the exterior angles is always 360°

#### Explanation:

The interior angle of a polygon can be found by:

$\frac{180 \left(n - 2\right)}{n}$ where $n$ is the number of sides

a) 4 sides
Interior angle is equal to:
$\frac{180 \left(4 - 2\right)}{4} = 90$

Therefore, exterior angle is equal to $180 - 90 = {90}^{\circ}$

The sum = 4 xx 90° = 360°

b) 5 sides
Interior angle is equal to:
$\frac{180 \left(5 - 2\right)}{5} = 108$

Therefore, exterior angle is equal to $180 - 108 = {72}^{\circ}$

The sum = 5 xx72° = 360°

c) 6 sides
Interior angle is equal to:
$\frac{180 \left(6 - 2\right)}{6} = 120$

Therefore, exterior angle is equal to $180 - 120 = {60}^{\circ}$

The sum = 6 xx 60° = 360°

d) n sides
Interior angle is equal to:
$\frac{180 \left(n - 2\right)}{n}$

So exterior angle is equal to $180 - \frac{180 \left(n - 2\right)}{n}$ which can be simplified

$180 - \frac{180 \left(n - 2\right)}{n}$

$= \frac{180 n - \left(180 \left(n - 2\right)\right)}{n}$

$= \frac{180 n - 180 n + 360}{n}$

exterior angle $= \frac{360}{n}$

The sum = 360/n xx n = 360°

Jul 29, 2018

${360}^{\circ}$

#### Explanation:

$\text{The sum of the exterior angles of any polygon is } {360}^{\circ}$