Regular Polygon Perimeter & Area Calculator
Find the perimeter and area of a regular polygon from the number of sides and the side length, plus the apothem and interior angle.
How it works
Perimeter = n · s · Area = ¼ · n · s² · cot(π/n) · Apothem = s / (2·tan(π/n))
A regular polygon has all sides and angles equal, so its perimeter is just the side length times the number of sides. The area is most cleanly written as one quarter of n·s² times the cotangent of π/n, which is the same as ½ · perimeter · apothem — the apothem being the perpendicular distance from the centre to a side. As the number of sides grows, the polygon approaches a circle.
Worked example
For a regular hexagon (n = 6) with side 10: perimeter = 60, apothem = 10 / (2·tan(30°)) = 10 / 1.1547 ≈ 8.6603, and area = ¼·6·100·cot(30°) = 150·1.7321 ≈ 259.8076 square units.
Sources: Regular polygon — Wikipedia
Frequently asked questions
What is the apothem?
The apothem is the perpendicular distance from the centre of the polygon to the midpoint of a side. It acts like the “radius” for area, since the area equals ½ · perimeter · apothem.
How do I find the interior angle?
Each interior angle of a regular n-gon is (n − 2)·180° / n. For a hexagon that is 4·180/6 = 120°.
Why does the polygon approach a circle?
As the number of sides increases with a fixed size, the perimeter approaches a circle’s circumference and the apothem approaches the radius, so the ½ · perimeter · apothem area tends to π·r².
Related calculators
This calculator is for educational purposes. Double-check important results before acting on them.