Regular Polygon Calculator
From the number of sides and side length, get perimeter, area, interior and exterior angles, apothem, and circumradius.
Not to scale
A regular hexagon with side 10: perimeter 60, interior angle 120°, apothem 8.6603, circumradius 10. Interior angles sum to 720°.
A regular polygon has equal sides and equal angles, so its measures follow from the number of sides n and side length s. A regular hexagon with side 10 has perimeter 60, area ≈ 259.8, an interior angle of 120°, an exterior angle of 60°, apothem ≈ 8.66, and circumradius 10.
What a regular polygon calculator does
A polygon is regular when every side is the same length and every interior angle is equal. That symmetry means two inputs — the number of sides n and the side length s — fix every other measure. The interior and exterior angles depend on n alone, while the perimeter, area, apothem, and circumradius scale with s.
π ÷ n is in radians; the interior angle is in degrees
Worked example
Take a regular hexagon, so n = 6 and side length s = 10:
- 1 Perimeter. P = n·s = 6 × 10 = 60.
- 2 Interior and exterior angles. Interior = (n − 2)·180 ÷ n = 4 × 180 ÷ 6 = 120°. Exterior = 360 ÷ n = 60°. They add to 180° at each vertex.
- 3 Apothem. a = s ÷ (2·tan(π ÷ n)) = 10 ÷ (2·tan30°) = 10 ÷ 1.1547 ≈ 8.66.
- 4 Area. A = n·s² ÷ (4·tan(π ÷ n)) = 600 ÷ (4·tan30°) = 600 ÷ 2.3094 ≈ 259.81. Equivalently A = ½·P·a = ½ × 60 × 8.66.
- 5 Circumradius. R = s ÷ (2·sin(π ÷ n)) = 10 ÷ (2·sin30°) = 10 ÷ 1 = 10.
Regular polygons at a glance
Interior angle depends only on n. The area factor is area ÷ s² = n ÷ (4·tan(π ÷ n)).
| Sides n | Name | Interior angle | Area factor (× s²) |
|---|---|---|---|
| 3 | Triangle | 60° | 0.4330 |
| 4 | Square | 90° | 1.0000 |
| 5 | Pentagon | 108° | 1.7205 |
| 6 | Hexagon | 120° | 2.5981 |
| 7 | Heptagon | ≈ 128.57° | 3.6339 |
| 8 | Octagon | 135° | 4.8284 |
Regular vs irregular, and the two radii
These formulas only hold for regular polygons. If the sides or angles differ, there is no single side length to plug in — an irregular polygon must be split into triangles (or handled with coordinates) to find its area.
The apothem is the perpendicular distance from the centre to the middle of a side; it is the radius of the inscribed circle and drives the area formula A = ½·P·a. The circumradius R reaches from the centre to a vertex, so it is the radius of the circle that passes through every corner. For any regular polygon a is always smaller than R, and the two meet only in the limit of infinitely many sides, where the polygon becomes a circle.