Regular Polygon Area (from Side Length)

One formula covers every regular polygon — pentagon, hexagon, octagon, and on toward the circle. Slice the n-gon into n identical triangles meeting at the centre; each has base $s$ and height equal to the apothem $s/(2\tan(\pi/n))$, and summing them gives $A = ns^2/(4\tan(\pi/n))$.

Two sanity checks live inside it. Set $n = 4$ and $\tan(\pi/4) = 1$, so the expression collapses to $s^2$ — the square, exactly. And as $n$ grows large the polygon becomes indistinguishable from a circle, which is precisely how Archimedes estimated π: by squeezing a circle between inscribed and circumscribed 96-gons.

The hexagon is the one nature chose. Of the three regular shapes that tile a plane without gaps — triangle, square, hexagon — the hexagon encloses the most area per unit of perimeter, which is why honeybees build in hexagons and use the least wax for the most honey.