How this calculator works
Join the centre to every vertex to split a convex regular polygon into equal isosceles triangles. Each triangle has one side as its base and the apothem as its perpendicular height. Multiplying its area by the number of sides gives the whole area. The circumradius reaches a vertex, while the apothem reaches a side midpoint; confusing them changes the answer. Reverse area mode reconstructs the dimensions from the requested area.
Formula
A = na²/[4 tan(π/n)] = np² tan(π/n) = nR² sin(2π/n)/2. Here a is side, p is apothem and R is circumradius. Also A = perimeter × p/2.
Worked example
A square with apothem 2 m has side 4 m. Each of its four central triangles has area 4 × 2/2 = 4 m², giving 16 m² in total. A regular hexagon with circumradius 2 m has side 2 m and area 6√3 ≈ 10.392305 m².
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Frequently asked questions
Can this calculate an irregular polygon?
No. The equal triangles depend on equal sides and angles. Use coordinates or split an irregular polygon into known shapes.
Why offer an apothem input?
It lets you work directly from the distance from the centre to an edge, often called the inradius. It also makes the perimeter-times-apothem method easy to follow.
Why is the area not perimeter times apothem?
That product would count each central triangle twice. Every triangle contributes half its base times its height.