How this calculator works
A regular polygon has all sides equal and all interior angles equal. Its centre connects to each vertex to form n identical isosceles triangles, which is the source of every formula here. The apothem is the distance from the centre to a side, and the circumradius reaches a vertex. Because the triangles are identical, the area is n times the area of one triangle, or half the perimeter times the apothem. The interior angle follows from dividing the total 180(n - 2) degrees among n vertices. Entering an area, perimeter or circumradius first recovers the side length, so any one of those measurements is enough.
Formula
With n sides of length a: perimeter = na; area = na²/(4 tan(π/n)); apothem = a/(2 tan(π/n)); circumradius = a/(2 sin(π/n)); interior angle = 180(n - 2)/n; exterior angle = 360/n; diagonals = n(n - 3)/2.
Worked example
A regular hexagon with side 5 m has perimeter 30 m and area 6 × 25/(4 tan 30°) ≈ 64.9519 m². Its apothem is 5/(2 tan 30°) ≈ 4.3301 m, its interior angle is 120°, and it has 9 diagonals.
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Frequently asked questions
How many sides can I enter?
The calculator accepts whole numbers from 3 to 100. Three sides describe a triangle, four a square, and larger values approach a circle.
What is the apothem?
The perpendicular distance from the centre to the midpoint of a side. It is the radius of the circle inscribed in the polygon.
Why does the exterior angle divide into 360 degrees?
Walking around a convex polygon turns through one exterior angle at each vertex. A full turn is 360 degrees, so each of n equal exterior angles is 360/n.
Can I use the circumradius instead of a side?
Yes. Choose the circumradius mode and enter the distance from the centre to a vertex. The side is then 2R sin(π/n).