Regular Polygon Area Calculator

Use $A = \frac{1}{4} n s^2 \cot\!\left(\frac{\pi}{n}\right)$, where $n$ is the number of sides and $s$ is the side length. Alternatively, $A = \frac{1}{2} \times \text{perimeter} \times \text{apothem}$, where the apothem is the distance from the center to the midpoint of a side.

The apothem is the perpendicular distance from the center of the polygon to the midpoint of any side. It equals $\frac{s}{2 \tan(\pi/n)}$, where $s$ is the side length and $n$ is the number of sides.

As the number of sides increases, the regular polygon approaches a circle. The area converges to $\pi r^2$, where $r$ is the circumradius. This is how Archimedes first approximated $\pi$ — by inscribing and circumscribing polygons with increasing numbers of sides.

A regular hexagon with side length $s$ has area $A = \frac{3\sqrt{3}}{2} s^2$. This equals 6 equilateral triangles, each with area $\frac{\sqrt{3}}{4} s^2$. A hexagon with 2 m sides has area $\approx 10.39$ m².

Each interior angle equals $\frac{(n - 2) \times 180^\circ}{n}$. For example: triangle = $60^\circ$, square = $90^\circ$, pentagon = $108^\circ$, hexagon = $120^\circ$, octagon = $135^\circ$. The angles approach $180^\circ$ as $n$ increases.