Regular Pentagon & Hexagon Area Calculator
Find the area, apothem and perimeter of a regular pentagon or hexagon from its side length.
How it works
Area = (n · s² · cot(π/n)) / 4 · Apothem = s / (2 · tan(π/n)) · Perimeter = n · s
A regular polygon of n equal sides can be split into n identical isosceles triangles meeting at the centre. Summing their areas gives the compact formula (n · s² · cot(π/n)) / 4. The apothem is the perpendicular distance from the centre to a side — equivalently the inradius — and equals s / (2·tan(π/n)). For a hexagon this simplifies to the well-known (3√3/2)·s².
Worked example
A regular hexagon (n = 6) with side 6: cot(π/6) = √3, so area = (6 · 36 · √3) / 4 = 54√3 ≈ 93.53 units²; apothem = 6 / (2·tan30°) = 3√3 ≈ 5.196 units; perimeter = 36 units. A pentagon (n = 5) with the same side gives ≈ 61.94 units².
Sources: Regular polygon — Wolfram MathWorld
Frequently asked questions
What is the apothem?
The apothem is the shortest distance from the polygon’s centre to the middle of any side. It is also the radius of the largest circle that fits inside the polygon (the inradius), and it makes the area equal to ½ × perimeter × apothem.
Why does a hexagon have such a clean formula?
A regular hexagon splits into six equilateral triangles, each of area (√3/4)s². Six of them give (3√3/2)s², which is why hexagons tile so efficiently and appear in honeycombs.
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