Regular N‑gon Calculator

Compute side length, circumradius, inradius, area, perimeter, interior angle & more. Instant real‑time update with interactive diagram.

General formulas (n = number of sides): Interior angle = (n‑2)·180°/n , side s = 2R·sin(π/n) , inradius r = R·cos(π/n) , area A = ¼·n·s²·cot(π/n)

Enter integer ≥ 3
Positive number
Common:
triangle
square
pentagon
hexagon
octagon
decagon
dodecagon
Polygon properties
Sides (n)
5
Side length (s)
1.000
Perimeter (P)
5.000
Area (A)
1.720
Circumradius (R)
0.851
Inradius (r)
0.688
Interior angle
108.0°
Central angle
72.0°
polygon
circumcircle (R)
incircle (r)
vertices

Understanding Regular Polygons

A regular polygon (or n‑gon) has all sides equal and all interior angles equal. It is both cyclic (all vertices lie on a common circle – circumcircle) and tangential (all sides touch an inner circle – incircle).

Key formulas (n sides, side s, circumradius R, inradius r, area A):

  • Interior angle = 180° × (n‑2) / n
  • Central angle = 360° / n
  • s = 2 R sin(π/n)
  • r = R cos(π/n) = ½ s cot(π/n)
  • A = ½ n R² sin(2π/n) = ¼ n s² cot(π/n) = n r² tan(π/n)
  • P = n s

Why “N‑gon”?

Greek prefix: tri‑(3), tetra‑(4), penta‑(5), hexa‑(6), hepta‑(7), octa‑(8), ennea‑(9), deca‑(10), etc. For higher numbers we simply say “n‑gon” (e.g., 17‑gon).

Real‑world applications

  • Architecture: floor plans, honeycomb (hexagons), domes.
  • Chemistry: cyclic molecules (benzene ring – hexagon).
  • Games & design: dice, board games, tiles.
  • Mathematics: construction of regular polygons, trigonometry.

Frequently Asked Questions

A polygon that is both equiangular (all angles equal) and equilateral (all sides equal). Examples: equilateral triangle, square, regular pentagon.

No, the number of sides n must be an integer ≥ 3. The calculator rounds it to the nearest integer and shows a warning if needed.

Circumradius (R) is the radius of the circle that passes through all vertices. Inradius (r) is the radius of the circle inscribed inside the polygon, tangent to every side.

Depending on the known input we use the most stable formula: e.g., from side: A = ¼ n s² cot(π/n); from circumradius: A = ½ n R² sin(2π/n).