Polygon Interior & Exterior Angle Calculator
Compute the interior and exterior angles and the total angle sum of any regular polygon from its number of sides.
How it works
Interior angle sum = (n − 2)·180° · Each interior = (n − 2)·180° ÷ n · Each exterior = 360° ÷ n
Any polygon with n sides can be cut into n − 2 triangles, and each triangle’s angles sum to 180°, so the interior angles total (n − 2)·180°. In a regular polygon those angles are all equal, so each is that sum divided by n. The exterior angle at each vertex is what remains to a straight line, and the exterior angles of any convex polygon always add up to exactly 360°.
Worked example
For a pentagon (n = 5): interior sum = (5 − 2)·180° = 540°, so each interior angle = 540 ÷ 5 = 108°. Each exterior angle = 360 ÷ 5 = 72°, and 108° + 72° = 180° as expected.
Frequently asked questions
Why do exterior angles always sum to 360°?
Walking once around the boundary of any convex polygon, you turn through each exterior angle and end up facing your original direction — a full 360° turn. That is true regardless of how many sides the polygon has.
Does this apply to irregular polygons?
The interior-angle sum (n − 2)·180° holds for any simple polygon, regular or not. The "each angle" figures here assume a regular polygon where all angles are equal.
What is the interior angle of a regular hexagon?
With n = 6, the sum is (6 − 2)·180° = 720°, so each interior angle is 720 ÷ 6 = 120°, and each exterior angle is 60°.
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This calculator is for educational purposes. Double-check important results before acting on them.