# Exterior Angles of a Polygon

How to solve the exterior angles of a polygon problems: formula, proof, examples, and their solutions.

## Formula

The sum of the measures of the exterior angles of an *n*-gon is 360.

## Proof

For an *n*-gon,

there are *n* of interior angles (purple)

and *n* of exterior angles (blue).

And there are *n* of linear pairs. (purple & blue)

Sum of interior angles: 180(*n* - 2)

Sum of exterior angles: (sum) ← wanted

Sum of the linear pairs: 180*n*

So 180(*n* - 2) + (sum) = 180*n*.

Interior angles of a polygon

## Example 1

The given angles are the exterior angles of a quadrilateral.

So (3*x* + 40) + (2*x* + 10) + (120) + (90) = 360.

## Example 2

A regular pentagon has 5 congruent sides

and 5 congruent angles.

So the measure of an exterior angle

of a regular pentagon is 360/5 = 72.

## Example 3

The regular polygon has *n* congruent sides

and *n* congruent angles.

And the measure of an interior angle

of the regular polygon is 60.

So 360/*n* = 60.