Hexagon Area Calculator

Formula

A = (3√3 / 2) × l² ≈ 2.598 × l²

Area of a regular hexagon

A regular hexagon has six equal sides, and each of its interior angles measures 120°. With a side length of L, the area works out to A = (3√3/2)·L² ≈ 2.598·L². Plug in L = 10 m and you get 2.598 · 100 ≈ 259.81 m². Here's a nice way to see why: split the hexagon into six equilateral triangles that all meet at the centre. Each one has area (√3/4)·L², and six of those add right back up to the same formula.

The longest diagonal, running from a vertex to the one across from it, comes to 2L, and that's also the diameter of the circumscribed circle. As for the apothem, a = (√3/2)·L ≈ 0.866·L.

Applications: from honeycombs to telecom cells

Among all the polygons that tile a flat surface, the regular hexagon is the only one that does it with the least possible perimeter for a given area. That's the honeycomb conjecture, which Thomas Hales finally proved in 1999. It's the reason bees build hexagonal combs (least wax, most honey), the reason graphene and snowflakes settle into hexagonal symmetry, and the reason cellular networks model their coverage as hexagons, with each base station handling a hex cell that barely overlaps its neighbours. The same logic is why hex tiles turn up on bathroom floors and in games like Catan.

FAQ

Why hexagons in beehives? Of all the shapes that tile a plane, the hexagon needs the least perimeter — and so the least wax — for each unit of area it encloses. Squares and triangles tile just fine, but they waste material doing it.

Perimeter of a regular hexagon? Six equal sides, so P = 6·L. With L = 10, that's P = 60 m.

How does it relate to the circle? The circumscribed circle has radius L, and the inscribed circle has radius equal to the apothem, (√3/2)·L. The hexagon itself fills about 82.7% of the area of that circumscribed circle.

Why use hex cells in mobile networks? Hexagonal cells blanket the plane with no gaps, and they keep every interior point about the same distance from the nearest base station boundary, which keeps handoff zones small. For a given number of antennas, that beats laying things out as squares or triangles.