Angles in polygons

A polygon with n sides can be split into n − 2 triangles from one corner, so its interior angles add up to (n − 2) × 180°. The exterior angles of any polygon add up to 360°.

Key facts

  • Interior angles of an n-sided polygon add to (n − 2) × 180°.
  • Exterior angles always add to 360°.
  • Interior + exterior at the same corner = 180°.
  • Regular polygon: every exterior angle is 360 ÷ n.

Worked examples

Example 1

Find the sum of the interior angles of a pentagon.

  1. From one corner a pentagon splits into 3 triangles. 5 − 2 = 3
  2. Each triangle has 180°. 3 × 180 = 540°

Answer: 540°

Example 2

Find each interior angle of a regular hexagon.

  1. Exterior angles add to 360. 360 ÷ 6 = 60
  2. Interior and exterior make a straight line. 180 − 60 = 120°

Answer: 120°