Polygon Area Calculator

Half the perimeter times the apothem, where the apothem is the distance from the centre to the middle of a side. It works for any regular polygon.

The perpendicular distance from the centre to a side, which is the polygon's equivalent of a circle's radius for area purposes. The circumradius, to a vertex, is always longer.

(n − 2) × 180 degrees. A triangle gives 180, a quadrilateral 360, a pentagon 540. It comes from splitting the polygon into n − 2 triangles.

Because walking once around the shape turns you through one full rotation regardless of how many corners there are. That is why each exterior angle of a regular polygon is simply 360 ÷ n.

The polygon approaches its circumscribed circle. Archimedes used exactly this to bound π between 96-sided polygons inscribed and circumscribed, two thousand years before calculus.

No — it assumes all sides and angles are equal. Irregular polygons need the shoelace formula from the vertex coordinates, or division into triangles.

A method for the area of any polygon from its vertex coordinates, cross-multiplying pairs and halving the total. It is called that because the multiplication pattern looks like lacing.

Because they tile a plane with the least total boundary per unit area of the three shapes that tile at all. Honeycomb uses the least wax for the most cell volume.

Only triangles, squares and hexagons. Their interior angles divide evenly into 360 degrees, which no other regular polygon's does.

One with an interior angle above 180 degrees, so at least one vertex points inward. The apothem method here does not apply — only convex regular polygons have a single apothem.

n(n − 3) ÷ 2. A hexagon has nine, a decagon thirty-five. Each vertex connects to every other except itself and its two neighbours, and each diagonal is then counted twice.

Above about twelve sides, most people simply say n-gon — a 17-gon, a 100-gon. The Greek-derived names exist and stop being useful long before the shapes do.