Interior and exterior angles
At each vertex of a polygon there is an interior angle (inside the shape) and an exterior angle (formed by extending one side). At every vertex the interior and exterior angles lie on a straight line, so they add up to 180°.
Key idea / Key idea
Sum of interior angles of an n-sided polygon = (n − 2) × 180°.
Sum of exterior angles of ANY polygon = 360°.
At each vertex: interior + exterior = 180°.
Sum of interior angles
Any polygon can be split into triangles from one vertex. An n-sided polygon splits into (n − 2) triangles, and each triangle has angles summing to 180°.
Worked example / Worked example
Find the sum of interior angles of a hexagon (n = 6).
Sum = (6 − 2) × 180° = 4 × 180° = 720°.
Angles of a regular polygon
In a regular polygon all angles are equal, so we can divide the totals by n. Each exterior angle = 360° ÷ n, and each interior angle = 180° − exterior angle.
Worked example / Worked example
Find each interior angle of a regular pentagon (n = 5).
Each exterior angle = 360° ÷ 5 = 72°.
Each interior angle = 180° − 72° = 108°.
Check: 5 × 108° = 540° = (5 − 2) × 180°. Correct.
Remember / Remember
- Interior sum = (n − 2) × 180°.
- Exterior sum = 360° always.
- Interior + exterior = 180° at each vertex.
- Regular polygon: exterior angle = 360° ÷ n.