Sum of Angles in a Polygon
n sides → (n−2)·180°. A regular polygon also shows a single interior angle.
How do you find the interior angle of a regular polygon? Enter the number of sides n below, and the calculator first works out the sum of the interior angles of the polygon — using the formula (n−2)·180° — then divides it equally between all the sides to get the size of a single interior angle. For example, in a regular pentagon (n=5) the angle sum is 540°, so each interior angle equals 540/5 = 108°.
How to use it
Sum of interior angles in an n-sided polygon = (n−2)·180°. Why? Because any simple polygon can be split into (n−2) triangles (using diagonals from one vertex), and the angle sum in every triangle is 180°.
Regular polygon = a polygon where all the sides and all the angles are equal. In a regular polygon, each interior angle equals the angle sum divided by n.
- Equilateral triangle (n=3): sum (3−2)·180 = 180°. Each angle: 180/3 = 60°.
- Square (n=4): sum (4−2)·180 = 360°. Each angle: 360/4 = 90°.
- Regular hexagon (n=6): sum (6−2)·180 = 720°. Each angle: 720/6 = 120°.
Common mistake: confusing the interior angle with the exterior angle. The exterior angle of a regular polygon is always 360 ÷ n, and the sum of the exterior angles is always 360° — no matter how many sides there are!
💼 Real-life uses: architecture, tiling (tessellation), coin design, roundabouts and road layouts, load-bearing structures (hexagons in a honeycomb) and more.
Related tools
- Perimeter and Area — area and perimeter calculations for various shapes, including regular polygons.
- Volumes — the volume of 3D solids, some based on polygons (prisms).
- Pythagoras' Theorem — for finding side lengths in a right-angled triangle.
✍️ Written by the MathsUK teamChecked against the National Curriculum and GCSE specifications