How To Calculate The Area Of A Hexagon

A regular hexagon has six equal sides and six equal angles of 120 degrees. The word regular is essential, because an irregular hexagon has no single formula and must be split into triangles and added up.

The area formula comes from six equilateral triangles. Draw lines from the centre to each vertex and the hexagon falls into six equilateral triangles whose side is the hexagon side, and each has area the square root of three over four times s squared.

Six of those triangles give one and a half times the square root of three times s squared, which is about 2.598 times s squared. This constant is worth remembering because it converts a side length straight into an area.

The apothem is the perpendicular distance from the centre to the middle of a side, and for a hexagon it is the square root of three over two times the side. With a side of 6 the apothem is 5.1962, slightly less than the side itself.

The general polygon formula, half the perimeter times the apothem, works for any regular polygon and gives the same answer. Half of 36 times 5.1962 is 93.5307, which matches the triangle method exactly.

The circumradius equals the side length. A circle drawn through all six vertices has radius s, so a hexagon with sides of 6 fits inside a circle of radius 6, and that circle has an area of 113.0973.

The hexagon tiles the plane perfectly. Six hexagons meet at every vertex with no gaps, which is why honeycomb, paving and hexagonal grids all use the shape and why bees are such efficient builders.

It is the most efficient shape for enclosing area with a given perimeter. Among shapes that tile, the hexagon needs the least wall for the most floor, which is why it appears in everything from city planning studies to heat exchanger design.

A cube viewed corner-on shows a regular hexagon. The shadow of a cube lit along a body diagonal is a hexagon, which is a handy way to see why sixfold symmetry appears in crystal structures.

Converting from the apothem is common in construction. If you know the distance across the flats, divide by 0.8660254 to get the side, then apply the area formula. Bolt heads and nuts are specified this way.

The interior angle of 120 degrees makes hexagons mechanically stable. Three struts meeting at a vertex share the load evenly, which is why hexagonal lattice structures are strong for their weight.

For a quick estimate, two and a half times the side squared is close enough for most field work. The exact constant is 2.5980762, and the shortcut is within about four percent.

Hexagonal grids appear in game design because a hexagon has six neighbours at equal distance, unlike the four of a square grid or the eight uneven neighbours of a triangle grid.

In optics, a hexagonal aperture produces diffraction spikes arranged with sixfold symmetry, which is why some telescope images show a six pointed star around bright stars.

Packaging engineers use hexagons for tubes and containers because the shape nests without wasted space between units, which is exactly why honeycomb cardboard exists.

When specifying a hexagonal window or panel, the dimension usually quoted is across the flats, so halve it and divide by 0.8660254 to recover the side before applying the area formula.

The constant can be derived a second way. A regular hexagon is half of a regular hexagram, and counting those triangles gives the same 2.5980762 factor that the equilateral triangle route produces.