Regular Polygon Calculator

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.