Hexagon Calculator

How to use this calculator

1. Pick which dimension you haveA regular hexagon is fully determined by any single measurement, so whichever one you can measure is enough to find all the rest.
2. Be clear which diagonal you meanAcross the flats and across the corners are different measurements and differ by about 15 per cent. Spanner sizes and bar stock are quoted across the flats; the corners are what has to clear an obstruction.
3. Units carry throughThe calculator works in whatever unit you enter. Enter millimetres and you get millimetres and square millimetres back.
4. Read the whole breakdownAll six dimensions are returned at once, which is usually faster than running the calculator repeatedly for each one.

How the calculation works

A regular hexagon has a property that makes all its arithmetic straightforward: drawing lines from the centre to each vertex divides it into six equilateral triangles. Because those triangles are equilateral, the distance from the centre to any corner — the circumradius — is exactly the side length.

Everything else follows from that. The apothem, the perpendicular distance from the centre to the middle of a flat, is the height of one of those equilateral triangles: s√3 ÷ 2, or about 0.866s.

Across the flats is twice the apothem, so s√3 ≈ 1.732s. Across the corners is twice the circumradius, so exactly 2s.

The area is six equilateral triangles, each of area (√3 ÷ 4)s², giving (3√3 ÷ 2)s² ≈ 2.598s².

That single constant, 2.598, is worth remembering. It means a hexagon has about 2.6 times the area of a square drawn on its side length, and about 82.7 per cent of the area of its circumscribed circle.

The relationships

Working backwards from across the flats is the most common practical need, because that is how hexagon bar, nuts and spanner sizes are specified. Divide the across-flats dimension by 1.7321 to get the side, and everything else follows.

Worked example

A hexagon with a 10 mm side.

Perimeter is 60 mm. Across the corners is 20 mm. Across the flats is 10 × 1.7321 = 17.32 mm. The apothem is 8.66 mm. The area is 2.5981 × 100 = 259.81 mm².

Now work it the other way, which is the realistic case. A 24 mm spanner fits a nut measuring 24 mm across the flats. The side is 24 ÷ 1.7321 = 13.856 mm, and across the corners is 27.71 mm.

That last figure matters practically: a socket has to clear 27.71 mm, not 24 mm, so a recess drilled at 25 mm will not accept the nut however carefully it is centred. Confusing across-flats with across-corners is the single most common hexagon error in workshop and design work, and it costs a remade part rather than a wrong answer on paper.

Why hexagons turn up everywhere

The hexagon is the only regular polygon other than the triangle and square that tiles a plane with no gaps, and among those three it encloses the most area for a given total edge length. That combination is why it appears so often in both nature and engineering.

Honeycomb is the standard example, and the geometry is genuinely the reason: bees building hexagonal cells use less wax per unit of storage than any other tiling shape would require. The same logic drives hexagonal cores in lightweight composite panels, where the aim is maximum stiffness for minimum material.

Fasteners are hexagonal for a different reason. Six flats give a spanner six engagement positions at 60-degree intervals, which is enough purchase to apply torque without rounding the corners, while still allowing the tool to be repositioned in a confined space. Square heads give only four positions and round more easily; twelve-point fittings give more positions and less contact area per flat.

Hexagonal tiling also appears in graphics and games because a hexagonal grid gives six equidistant neighbours, unlike a square grid where the four diagonal neighbours are further away than the four orthogonal ones. That uniformity makes movement and distance calculations behave consistently in every direction.

Assumptions and limitations

• Applies to regular hexagons only — all sides equal and all interior angles 120°
• Irregular hexagons must be split into triangles and calculated separately
• Works in whatever unit you enter and cannot detect mixed units
• Across the flats and across the corners are different measurements — check which your source means
• Results are rounded for display; use full precision for onward calculations
• Assumes a two-dimensional shape — hexagonal prisms need the length as well

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