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

Question

directrix · Answer

The sum of the exterior angles, one per vertex, of a regular polygon is 360. If one exterior angle is 72, then the number of sides of the polygon is 360/72 = 5. The regular polygon is a regular pentagon. 

If the exterior angle measures 72, its adjacent interior angle is 180 - 72 = 108. The sum of the interior angles is 108 times 5 = 540 degrees.