# What is the sum of the measures of the interior angles of a regular polygon where a single exterior angle measures 72°?

Jan 28, 2016

The sum of the interior angles would be ${540}^{o}$

#### Explanation:

If the regular polygon has an exterior angle of ${72}^{o}$ this means there would be 5 sides to the polygon and it would be a pentagon.

The exterior measures of a polygon must add up to ${360}^{o}$, therefore ${360}^{o} / {72}^{o}$ would mean 5 exterior angles, five interior angles and therefore 5 sides to the polygon. (see diagram below)

The exterior angle and the interior angle of any polygon are supplementary and must add up to ${180}^{o}$.

${180}^{o} - {72}^{o} = {108}^{o}$
Each interior angle would therefore be ${108}^{o}$

Since there are 5 angle in a pentagon $5 x {108}^{o} = {540}^{o}$