Regular Polygon Area Calculator
The area of any regular polygon from its side length and number of sides — triangle to decagon and beyond.
A regular polygon splits into n identical triangles meeting at the center.
How the regular polygon area calculator works
A regular polygon splits into n identical triangles meeting at the center. Each has base a and height equal to the apothem, a ÷ (2·tan(π/n)), which gives the area n·a² ÷ (4·tan(π/n)). As n grows the shape approaches a circle: a hundred-sided polygon of the same perimeter is within a hundredth of a per cent of the circle's area.
Formula: square a²; rectangle ab; triangle ½bh; circle πr²; parallelogram bh; trapezoid ½(a+b)h; ellipse πab
Worked examples
| Inputs | Area | Note |
|---|---|---|
| A hexagon of side 8 | 166.2769 | 166.28 |
| A pentagon of side 10 | 172.0477 | 172.05 |
| A 100-sided polygon | 795.5129 | almost a circle |
FAQFrequently asked questions
What makes a polygon regular?
All sides the same length and all angles equal.
What is the apothem?
The perpendicular distance from the center to the middle of a side — the height of each of the n triangles.
What is the formula?
n·a² ÷ (4·tan(π/n)), where a is the side and n the number of sides.
What are the interior angles?
(n − 2) × 180° ÷ n — 60° for a triangle, 90° for a square, 120° for a hexagon.
Why do hexagons tile so well?
Because 120° divides 360° exactly three ways, so three hexagons meet at a point with no gap — which is why honeycomb is hexagonal.
Where these figures come from
- NIST Digital Library of Mathematical Functions — reference definitions for elementary and special functions
- Wolfram MathWorld — definitions and formulas for every topic on this page
- NIST/SEMATECH e-Handbook of Statistical Methods — the statistical formulas (mean, variance, z, confidence intervals, sample size)
- Common Core State Standards — Mathematics — the terms and methods taught in US schools
Last checked: September 2026. Formulas are fixed by mathematics and do not change with tax years or regulations.