Angle sums in polygons
The interior angles of any polygon add up to a fixed total that depends only on the number of sides n. Splitting the polygon into triangles gives the formula below.
Key idea
Sum of interior angles = (n - 2) × 180°
Sum of exterior angles = 360° for any polygon.
For a regular polygon, each exterior angle = 360° / n.
Regular polygons
In a regular polygon all sides and all angles are equal. Each interior angle equals the total interior angle sum divided by n, and it is also 180° minus the exterior angle.
Worked example
Find each interior angle of a regular hexagon (n = 6).
Sum = (6 - 2) × 180° = 4 × 180° = 720°.
Each interior angle = 720° ÷ 6 = 120°.
Check with exterior angles: 360° ÷ 6 = 60°, and 180° - 60° = 120°. ✓
Using the exterior angle
To find the number of sides of a regular polygon from one exterior angle, divide 360° by that angle. Interior and exterior angles at each vertex always add to 180° because they lie on a straight line.
Remember
- Angles on a straight line add to 180°; angles round a point add to 360°.
- Triangle: 180°; quadrilateral: 360°; pentagon: 540°.
- Exterior angles of any polygon always total 360°.