Polygon angles

In a regular polygon all n interior angles are equal, so each one is (n − 2) × 180° ÷ n. A regular pentagon has 540° ÷ 5 = 108° at each vertex. At each vertex the exterior angle is the angle made by turning the extended side to the next side, and the interior and exterior angles together make a straight line: 108° + 72° = 180°. In the picture the exterior angle is 72°. The exterior angles of a regular polygon are equal, and they add up to 360° (next lesson), so each one is 360° ÷ n. For n = 5: 360° ÷ 5 = 72° and 180° − 72° = 108°. This gives the quickest way to find n: if an exterior angle is 24°, then n = 360 ÷ 24 = 15. If an interior angle is 150°, the exterior angle is 30° and n = 360 ÷ 30 = 12. Check that n comes out whole: an interior angle of 100° would give 360 ÷ 80 = 4.5, which is impossible.