Polygon Angle Calculator — Interior & Exterior Angles
Enter the number of sides n to find every angle property of a regular polygon: each interior angle, each exterior angle, the sum of interior angles, the central angle, and the number of diagonals.
Each interior angle of a regular Hexagon
- 1
Sum of interior angles
(6 − 2) × 180° = 720°Any n-sided polygon can be divided into (n − 2) triangles, each contributing 180°. - 2
Each interior angle
720° ÷ 6 = 120°
How does this calculator work?
For a regular n-sided polygon: interior angle = (n − 2) × 180° / n, exterior angle = 360° / n, and the sum of all interior angles = (n − 2) × 180°. For a triangle (n = 3) this gives 60° each; for a square (n = 4) 90° each; for a regular hexagon (n = 6) 120° each. The total of all exterior angles is always 360°.
Formula
How this is calculated
A regular polygon has n equal sides and n equal angles. Dividing it into triangles from one vertex shows that any n-sided polygon can be split into (n − 2) triangles. Since each triangle's angles sum to 180°, the total interior angle sum is (n − 2) × 180°. Dividing by n gives each interior angle: (n − 2) × 180° / n.
The exterior angle is the supplement of the interior angle and equals 360° / n. This follows from the fact that walking around any convex polygon and turning at each vertex rotates exactly once (360° total), so each exterior turn is 360° / n. Interior plus exterior angles always add to 180° at each vertex.
The central angle is the angle at the polygon's centre subtended by one side, which equals 360° / n (same as the exterior angle for a regular polygon). The number of diagonals is n(n − 3) / 2, derived by counting line segments between non-adjacent vertices. These formulas assume a convex regular polygon; irregular or concave polygons have the same interior angle sum but their individual angles vary.
Frequently asked questions
As a regular polygon gains more and more sides, it approaches a circle and its interior angles approach 180°. A circle can be thought of as a limit of regular polygons with infinitely many infinitely short sides.
The sum of interior angles, (n − 2) × 180°, holds for any simple (non-self-intersecting) polygon, regular or not. The individual angle formula (n − 2) × 180° / n and the diagonal formula n(n − 3) / 2 are specific to regular polygons where all sides and angles are equal.
The interior angle is the angle inside the polygon at a vertex, between two adjacent sides. The exterior angle is the angle you turn through when walking along the boundary; it is the supplement of the interior angle (they add to 180° for a convex polygon). The sum of all exterior angles of any convex polygon is always 360°.
Also known as
TG we-Calculate Editorial Team. (2026). Polygon Angle Calculator — Interior & Exterior Angles [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/polygon-angle-calculator
TG we-Calculate Editorial Team. "Polygon Angle Calculator — Interior & Exterior Angles." TG we-Calculate. 2026. https://we-calculate.com/calculator/polygon-angle-calculator.
TG we-Calculate Editorial Team, "Polygon Angle Calculator — Interior & Exterior Angles," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/polygon-angle-calculator
@misc{wecalculate_polygon_angle_calculator, title = {Polygon Angle Calculator — Interior & Exterior Angles}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/polygon-angle-calculator}}, year = {2026}, note = {TG we-Calculate} }
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