Regular Polygon Calculator (any number of sides)
Calculate area, perimeter, apothem, and circumradius of any regular n-sided polygon from its side length.
Works from triangle (n=3) to 100+.
A regular n-sided polygon has n equal sides and n equal angles. Triangle, square, pentagon, hexagon, heptagon, octagon, nonagon, decagon — all the way up to “very many sides” which approaches a circle.
Perimeter: P = n × s
Interior angle (per vertex): angle = (n − 2) × 180° / n
Apothem (inradius — center to mid-side): r = s / (2 × tan(π/n))
Circumradius (center to vertex): R = s / (2 × sin(π/n))
Area: A = (n × s² / 4) × cot(π/n) = (1/2) × P × r
The area formula has a clean interpretation: a regular polygon’s area equals half its perimeter times its apothem. This generalizes the triangle formula (½ × base × height) to any regular polygon.
Reference values for common polygons (s = 1 unit):
| Sides | Name | Area | Perimeter | Apothem | Interior angle |
|---|---|---|---|---|---|
| 3 | Triangle (equilateral) | 0.4330 | 3.000 | 0.289 | 60° |
| 4 | Square | 1.000 | 4.000 | 0.500 | 90° |
| 5 | Pentagon | 1.720 | 5.000 | 0.688 | 108° |
| 6 | Hexagon | 2.598 | 6.000 | 0.866 | 120° |
| 7 | Heptagon | 3.634 | 7.000 | 1.038 | 128.57° |
| 8 | Octagon | 4.828 | 8.000 | 1.207 | 135° |
| 9 | Nonagon | 6.182 | 9.000 | 1.374 | 140° |
| 10 | Decagon | 7.694 | 10.000 | 1.539 | 144° |
| 12 | Dodecagon | 11.196 | 12.000 | 1.866 | 150° |
| 100 | Centagon | 795.51 | 100.000 | 15.91 | 176.4° |
As n increases, the polygon approaches a circle. For very large n with side length 1, the circumscribed circle has radius ≈ s × n / (2π).
Where this matters in real life:
- Gazebo and pavilion floors. Most are octagonal, hexagonal, or pentagonal. The same formulas size them all.
- Architectural cupolas and rooftop ornaments. Often heptagonal (7 sides) or nonagonal (9 sides) for visual variety.
- Coin design. Many world coins are heptagonal (Britain’s 50p), pentagonal (Australian commemorative), or polygonal for tactile distinction.
- Geodesic dome panels. A geodesic dome’s panels are triangles, but the projected ground footprint is often a regular polygon (10, 12, 15 sides).
- Stop signs are octagonal, Yield signs are triangular, Children Crossing signs are pentagon-shaped in some countries.
Worked example — building a heptagonal (7-sided) gazebo:
You want a unique 7-sided gazebo with 4-ft sides. Perimeter = 28 ft of edge. Area = 3.634 × 16 = 58.1 sq ft of floor space. Apothem = 1.038 × 4 = 4.15 ft (the distance from center to the middle of each wall). Interior angle = 128.57° (each corner is slightly more “open” than the 120° of a hexagon).
Worked example — designing a polygonal flower bed:
You want a 9-sided (nonagonal) decorative bed with 3-ft sides. P = 27 ft of bed border. A = 6.182 × 9 = 55.6 sq ft. Apothem = 1.374 × 3 = 4.12 ft (so a central plant fits within 4 ft of any edge).
Interesting limit: as n grows, the polygon “becomes” a circle. For n = 1000, the area is 999.97% of the inscribed circle area — visually indistinguishable. This is why ancient methods of computing π (like Archimedes’ polygon method) inscribed and circumscribed polygons with many sides and squeezed the answer between the two.