Polygon Interior Angle Calculator
Work out the interior and exterior angles of a polygon from its number of sides. Enter how many sides the shape has and get the total interior-angle sum instantly, plus each interior and exterior angle for a regular polygon.
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How to use this calculator
- Enter the number of sides (n) — a whole number of 3 or more, e.g. 6 for a hexagon.
- Read the sum of interior angles, which holds for any polygon with that many sides.
- Use the per-angle results when the shape is regular (all sides and angles equal).
How it works
The interior angles of any simple polygon with n sides always add up to (n − 2) × 180°, because the polygon can be split into (n − 2) triangles that each contribute 180°. For a regular polygon every angle is identical, so each interior angle is that sum divided by n, and each exterior angle is 360° ÷ n (the exterior angles of any convex polygon always total 360°). Interior and exterior angles at each vertex are supplementary, adding to 180°.
Worked example
Worked example. A regular hexagon has n = 6. The sum of interior angles is (6 − 2) × 180° = 720°. Each interior angle is 720° ÷ 6 = 120°, and each exterior angle is 360° ÷ 6 = 60°. Notice 120° + 60° = 180°, confirming they are supplementary.
Common mistakes
- Applying the per-angle result to an irregular polygon — the angle sum is the same, but individual angles vary unless the shape is regular.
- Entering fewer than 3 sides; a polygon must have at least three sides to enclose an area.
- Confusing the interior-angle sum with a single angle — 720° is the total for a hexagon, not one of its corners.
Frequently asked questions
What is the formula for the sum of interior angles?
Sum = (n − 2) × 180°, where n is the number of sides. This works for any simple polygon, convex or concave, regular or irregular.
Do the per-angle results work for irregular polygons?
No. The interior-angle sum is still (n − 2) × 180°, but 'each interior angle' and 'each exterior angle' only apply when the polygon is regular, meaning all sides and angles are equal.
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Tip: Enter any known values to calculate the remaining results.
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