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Regular Polygon Calculator

Find a regular polygon's interior and exterior angles, diagonal count, perimeter, apothem, and area from its number of sides and side length.

Interior angle
120°
Exterior angle
60°
Sum of interior angles
720°
Diagonals
9
Perimeter
60
Apothem
8.6603
Area
259.8076

How it works

Any simple n-sided polygon can be split into (n−2) triangles from one vertex, and triangles always sum to 180°, so the interior angles of any polygon sum to (n−2)×180°. A regular polygon splits that total evenly across its n equal vertices, giving an interior angle of (n−2)×180°÷n. The exterior angle is what's left from a straight 180° turn at each vertex, which simplifies to 360°÷n — and all n exterior angles always sum to exactly 360°, one full turn around the shape.

The number of diagonals comes from counting every possible vertex-to-vertex line (n choose 2, or n(n−1)/2 pairs) and subtracting the n edges themselves, which aren't diagonals: diagonals = n(n−3)/2.

For area, the apothem — the distance from the center to the midpoint of a side — is s ÷ (2×tan(π/n)), found from the right triangle formed by the center, a vertex, and a side's midpoint. Area is then ½ × perimeter × apothem, the same 'sum of n congruent triangles from the center' derivation used for any regular polygon, which reduces to the familiar ½ × base × height for each of those n triangles.

FAQ

Why do the exterior angles always add up to 360°, no matter how many sides?

Walking around any convex polygon's perimeter and turning by each exterior angle at every vertex brings you back to your starting direction after one full lap — that's a single 360° turn, regardless of how many sides or turns it took to get there.

Does this work for non-regular polygons too?

Only the sum of interior angles, (n−2)×180°, holds for any simple polygon regardless of shape. Everything else here — the individual interior/exterior angle, the apothem, and the area formula — assumes every side and angle is equal, which is what 'regular' means; an irregular polygon needs its own coordinates or side/angle measurements to solve.

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