Works for any regular n-gon — triangle, square, pentagon, hexagon, octagon, and beyond. Set the number of sides, tell the tool whether you know a side length, a radius, or the area, and it derives every other property: area, perimeter, circumradius, inradius (apothem), and the interior and exterior angles.
How it works
A regular polygon with n sides of length s has these exact relationships, which the tool uses:
- Perimeter = n × s
- Area = ¼ × n × s² × cot(π/n) — equivalently ½ × perimeter × apothem
- Circumradius (R) = s ÷ (2 × sin(π/n))
- Inradius / apothem (r) = s ÷ (2 × tan(π/n))
- Interior angle = (n − 2) × 180 ÷ n degrees
- Exterior angle = 360 ÷ n degrees
If you supply a radius or the area instead of the side, the tool inverts the matching formula to recover the side length, then computes everything else from it.
Example: regular hexagon
A regular hexagon (n = 6) with side 10:
- Perimeter = 6 × 10 = 60
- Area = ¼ × 6 × 100 × cot(30°) = 150 × √3 ≈ 259.81
- Circumradius = 10 ÷ (2 × sin 30°) = 10
- Apothem = 10 ÷ (2 × tan 30°) ≈ 8.66
- Interior angle = (6 − 2) × 180 ÷ 6 = 120°
- Exterior angle = 360 ÷ 6 = 60°
Notice that for a hexagon specifically, the circumradius equals the side length — a useful property when packing hexagons in tiling patterns.
Common polygon reference
| Sides (n) | Name | Interior angle | Exterior angle | Circumradius / side |
|---|---|---|---|---|
| 3 | Equilateral triangle | 60° | 120° | ≈ 0.577 |
| 4 | Square | 90° | 90° | ≈ 0.707 |
| 5 | Pentagon | 108° | 72° | ≈ 0.851 |
| 6 | Hexagon | 120° | 60° | 1.000 |
| 8 | Octagon | 135° | 45° | ≈ 1.307 |
| 12 | Dodecagon | 150° | 30° | ≈ 1.932 |
What the three radii mean in practice
Circumradius is the distance from centre to vertex. Use it when you need to fit the polygon inside a circle, or when you know the diagonal span and want to size it.
Inradius (apothem) is the distance from centre to the middle of a side — the radius of the inscribed circle that touches each side at its midpoint. It is the most useful measurement when packing polygons in tiles, since the apothem equals half the gap between polygon centres in a regular tiling.
Diagonal (for n ≥ 4) is the longest distance between two non-adjacent vertices. The tool computes this too: for a hexagon it equals 2R; for a square it equals R√2.
Edge cases and large n
As n grows large, a regular polygon approaches a circle: its interior angle approaches 180°, its apothem approaches its circumradius, and its area approaches π × R². A polygon with n = 360 is a useful approximation of a circle in design and fabrication contexts where curves are constructed from straight segments.
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