How to use it
- Enter the number of sides: 5 for a pentagon, 6 for a hexagon, 8 for an octagon, and so on.
- Choose what you know: the side length, the circumradius (center to a corner), the apothem (center to the middle of a side), the area or the perimeter.
- Enter the value. Lengths come out in the same unit, the area in that unit squared.
Formulas
For n sides of length s, with t = 180°/n:
- Apothem: a = s / (2 tan t)
- Circumradius: R = s / (2 sin t)
- Area: A = n·s·a / 2 = n·s² / (4 tan t)
- Perimeter: P = n·s
- Interior angle: (n − 2)·180° / n; exterior angle: 360° / n
- Diagonals: n(n − 3) / 2
From R the side is 2R sin t, from the apothem 2a tan t, from the area √(4A tan t / n).
Worked example
A regular hexagon with side 10: t = 30°, tan 30° = 0.57735. The apothem is 10 / (2 × 0.57735) = 8.6603, the circumradius 10 / (2 × 0.5) = 10, and the area 6 × 10 × 8.6603 / 2 = 259.8076. The perimeter is 60, each interior angle 120°, each exterior angle 60°, and there are 9 diagonals.
Tips
- Hexagon shortcut: the circumradius equals the side, and the area is 3√3/2 × s².
- Square check: with n = 4 and side 2 you get area 4 and apothem 1, a quick way to see the tool works as expected.
- Many sides: as n grows the polygon approaches its circumcircle; with n = 1000 the area is within 0.001 % of πR².