Area = half the perimeter times the apothem. A unit hexagon has perimeter 6 and apothem √3/2, so its area is 3√3/2 ≈ 2.598. The formula holds for every regular polygon, and it is the same one a circle uses with its circumference and radius.
Two radii, not one
A regular polygon has two distances from its centre, and mixing them up is the usual error.
The circumradius reaches a corner. It is the radius of the circle through all the vertices.
The apothem reaches the middle of a side, perpendicular to it. It is the radius of the circle that just touches every side from inside.
The apothem is always the shorter, and the area formula wants that one. Using the circumradius instead overstates the area, and by more the fewer sides there are.
A hexagon is the memorable exception in the other direction: its circumradius exactly equals its side length, which is why six circles pack round a seventh and why honeycomb works.
The angles
Three angles matter, and one of them makes the others easy.
The exterior angles always add to 360°, for every polygon regardless of how many sides it has. So each one is 360/n, and that is the quickest thing to compute.
The interior angle is then 180° minus that, or equivalently (n − 2)×180/n. Going the exterior route avoids the (n − 2) that people misremember.
The central angle — the angle at the centre subtended by one side — is also 360/n, the same number as the exterior angle. That is not a coincidence: the two are corresponding angles across a pair of parallel lines.
Interior angles reaching 180° is why only triangles, squares and hexagons tile the plane on their own — those are the only ones whose interior angle divides 360°.
Which ones can be drawn
With only a compass and an unmarked straightedge, some regular polygons can be constructed exactly and some cannot. The answer is one of the more surprising results in mathematics.
Gauss settled it in 1796, at nineteen: a regular n-gon is constructible exactly when the odd part of n is a product of distinct Fermat primes — 3, 5, 17, 257 and 65537, the only ones known.
So a triangle, square, pentagon, hexagon and 17-gon can all be drawn. A 7-gon cannot — 7 is not a Fermat prime. Nor can a 9-gon, because 9 is 3×3 and the primes have to be distinct.
Gauss was reportedly so pleased with the 17-gon that he asked for one on his gravestone. The stonemason declined, on the grounds that it would be indistinguishable from a circle.
“Not constructible” means not exactly, with those two tools. Every one of them can be drawn to any accuracy you like by other means, which is what this page does.
Towards a circle
Add sides and a regular polygon becomes indistinguishable from its circumscribed circle. This page reports what share of that circle it fills.
A triangle manages 41%. A hexagon 83%. A 20-gon 97%. By a hundred sides it is over 99.9%.
Archimedes turned exactly this into a method. By bounding a circle between inscribed and circumscribed 96-sided polygons he pinned π between 3⅝₁ and 3⅞, which is accurate to two decimal places and stood for centuries.
The area formula shows the limit directly. Half the perimeter times the apothem becomes half the circumference times the radius as the sides multiply — and ½ × 2πr × r is πr².
Where it gets used
Construction and making. Setting out a hexagonal deck, an octagonal gazebo or a polygonal planter needs the apothem to mark the sides and the circumradius to mark the corners.
Engineering. Nuts and bolt heads are hexagonal because six flats give a spanner a good grip while wasting little material. The across-flats measurement is twice the apothem.
Tiling and packing. Hexagons tile the plane with the least perimeter for a given area, which is why honeycomb, basalt columns and graphene all arrive at the same shape.
Games and graphics. Hexagonal grids avoid the diagonal-distance problem that square grids have, since every neighbour is the same distance away.
Sources and methodology
The formulas and the constructibility result are standard; these are the references.
Method. Every input route reduces to the side length first and then derives everything else from it, so the five accepted measurements cannot disagree with each other — the suite checks all five round-trip back to the same polygon for every side count from 3 to 24. Areas are separately verified against the shoelace formula applied to generated vertex coordinates, which is a genuinely independent route rather than the same formula rearranged. That engine is verified on every change against 114 hand-written assertions, including that every one of the five input routes reconstructs the same polygon, and that the areas match a shoelace calculation on generated vertices from 3 sides to 40. The count and the per-case breakdown are published on the formula verification page.
Read the guide
For a polygon that is not regular, the Polygon Area Calculator works from the corner coordinates instead.