What is a Regular Polygon?
A regular polygon is a 2D closed figure with all sides of equal length and all interior angles of equal measure. Common examples include equilateral triangles, squares, regular pentagons, hexagons, and octagons. This polygon calculator finds all essential properties of any regular n-sided polygon — area, perimeter, interior and exterior angles, circumradius (R), and inradius (r).
Regular Polygon Formulas
For a regular polygon with n sides and side length s:
| Property | Formula |
|---|---|
| Perimeter (P) | P = n × s |
| Area (A) | A = (n × s²) / (4 × tan(π/n)) |
| Interior angle (α) | α = (n−2) × 180° / n |
| Exterior angle (β) | β = 360° / n |
| Circumradius (R) | R = s / (2 × sin(π/n)) |
| Inradius (r) | r = s / (2 × tan(π/n)) |
Names of Regular Polygon Shapes
| Sides (n) | Name | Interior angle (α) | Exterior angle (β) |
|---|---|---|---|
| 3 | Equilateral Triangle | 60° | 120° |
| 4 | Square | 90° | 90° |
| 5 | Pentagon | 108° | 72° |
| 6 | Hexagon | 120° | 60° |
| 7 | Heptagon | ≈128.57° | ≈51.43° |
| 8 | Octagon | 135° | 45° |
| 9 | Nonagon | 140° | 40° |
| 10 | Decagon | 144° | 36° |
| 11 | Hendecagon | ≈147.27° | ≈32.73° |
| 12 | Dodecagon | 150° | 30° |
| n | n-gon | (n−2)×180°/n | 360°/n |
How to Use This Polygon Calculator
- Enter the number of sides (n ≥ 3). You can use any integer.
- Choose your input: side length, circumradius (R), or inradius (r).
- Enter the value and select your unit (metric or US imperial).
- Click Calculate to instantly get all polygon properties.
Circumradius vs Inradius
The circumradius (R) is the radius of the circumscribed circle — the circle that passes through all vertices of the polygon. The inradius (r) is the radius of the inscribed circle — the largest circle that fits inside the polygon, touching each side at its midpoint.
Metric and Imperial Units
This calculator supports both metric (mm, cm, m, km) and US imperial (in, ft, yd, mi) measurement systems. Area is always displayed in the square of the selected unit (e.g., cm²).