How to find a regular polygon's area
A regular polygon has equal sides and angles. Its area is A = (1/4) × n × s² × cot(π/n), where n is the number of sides and s is the side length. For a hexagon (n = 6) with side 4, A = (1/4) × 6 × 16 × cot(30°) = about 41.57 square units.
The formula comes from slicing the polygon into n identical triangles that meet at the centre. Each triangle has base s and a height called the apothem, and the cotangent term is exactly what converts the side length into that apothem. Side length times the number of sides also gives the distance round the outside, as the perimeter calculator does for any shape.
How to use this calculator
Enter the number of sides (3 or more) and the side length. The cotangent term (cot = 1 ÷ tan) accounts for the polygon's shape. The result is in square units of your side length.
Enter the side length, not the radius or the apothem — those are different measurements and produce different answers. Because the formula contains s², doubling the side quadruples the area, so a small input error is magnified. The side count must be a whole number of at least 3.
Common polygons
This formula covers every regular shape. For specific cases you can also use the dedicated pentagon area calculator or hexagon area calculator, which use the same maths in a simpler form.
As n grows the shape gets rounder and its area climbs towards that of a circle with the same perimeter — which is why circles enclose more area than any polygon of equal perimeter.
Frequently asked questions
- What is the formula for a regular polygon's area?
- A = (1/4) × n × s² × cot(π/n), where n is the number of sides and s is the side length.
- What is the area of a regular hexagon with side 4?
- About 41.57 square units, using n = 6 and s = 4.
- How many sides can I use?
- Any whole number from 3 upward — triangle, square, pentagon, hexagon, and beyond.
- What is cot in the formula?
- Cotangent, which equals 1 ÷ tangent. The calculator handles it for you.
- Does this work for irregular polygons?
- No — the formula assumes every side and every angle is equal. For an irregular shape, split it into triangles and add their areas, or use the shoelace formula on the vertex coordinates. Feeding an irregular shape's side length in here will overstate the area.