How to Find the Area of a Hexagon

Of all the regular polygons this is the friendliest, because it comes apart into shapes you already know. Draw the three main diagonals of a regular hexagon and you have six equilateral triangles, all identical, all with the same side as the hexagon itself. Work out one and multiply by six.

Six equilateral triangles, and why they fit

The reason the fan works so neatly is a quiet property of hexagons: the distance from the centre to a corner is exactly the side length. For our hexagon that is 6 in both cases. So each of the six triangles has two sides of 6 radiating from the centre and a third side of 6 along the edge, which makes it equilateral without anyone arranging it.

An equilateral triangle of side 6 holds √3 over 4 times 36, which is 15.59. Six of those is 93.53. Multiply the triangle formula by six and you get 6 times √3 a² over 4, which tidies to 3√3 a² over 2. The famous constant is nothing more than six triangles collected up.

Why hexagons are everywhere

Honeycomb, bolt heads, floor tiles, carbon rings, chicken wire. The hexagon turns up wherever something has to be repeated cheaply, and the reason is measurable rather than aesthetic.

Take three shapes that all hold 93.53 square units. The circle needs a boundary of 34.28, the hexagon 36.00, the square 38.68, and an equilateral triangle a wasteful 44.09. The circle wins outright, but circles cannot be packed without leaving gaps between them. Restrict yourself to shapes that tile the plane and the hexagon is the most economical there is, which is why a bee building in hexagons spends the least wax per cell.

A worked check you can repeat on paper

That last pair is a useful sanity check. Draw a circle through the six corners and the hexagon fills 82.7 percent of it, always, whatever the size. If your answer is not somewhere near four fifths of πa², something has gone wrong upstream.

Across the flats, or across the corners

Real hexagons are rarely labelled with their side length. A bolt head or a spanner is specified across the flats, and a drawing might give the distance across the corners instead. Both are easy to convert, and mixing them up is the main way this calculation goes wrong outside a classroom.

Measurement	Relationship to the side	For a side of 6
Across the corners	twice the side	12.00
Across the flats	side times √3	10.39
Apothem, centre to a flat	side times √3 over 2	5.196
Corner to centre	the side itself	6.00

So a 24 mm spanner fits a nut measuring 24 mm across the flats, which means a side of 24 over √3, or 13.86 mm, and a hexagonal face of 498.8 square millimetres. Feed the 24 straight in as a side and you get 1496 instead, three times too much.

Mistakes that catch people out

• Using a width as the side. Across the flats is 10.39 here and across the corners is 12, neither of which is the side of 6. This is the error that ruins bolt and tile calculations.
• Using the apothem as the side. The apothem of 5.196 is shorter than the side, so this understates the area. If you have the apothem, either convert it or use half perimeter times apothem instead.
• Rounding 2.598 to 2.6. Harmless once, but on a floor of 400 tiles it drifts. The constant is 3√3 over 2, so keep the surd if the answer feeds an order.
• Assuming any six sided shape works. This formula needs a regular hexagon, with all sides and angles equal. An irregular one has to be split into triangles or run through the shoelace.
• Forgetting it is six triangles, not four. A quick sketch settles it and costs nothing.

Try these

Common questions

What is the formula for the area of a hexagon?

3√3 times the side squared, over 2, which works out as 2.598 times the side squared. A side of 6 gives 93.53 square units. This applies to a regular hexagon, with all sides and angles equal.

Why is the constant 3√3 over 2?

Because a hexagon is six equilateral triangles, and each holds √3 over 4 times the side squared. Six of those is 6√3 over 4, which simplifies to 3√3 over 2.

How do you find the area from the width across the flats?

Divide that width by √3 to get the side, then use the formula. A nut 24 mm across the flats has a side of 13.86 mm and a face area of 498.8 square millimetres.

What is the apothem of a hexagon?

The distance from the centre to the middle of a side, which is the side times √3 over 2. For a side of 6 it is 5.196, and half the perimeter times the apothem gives the area directly.

How do you find the area of an irregular hexagon?

This formula will not do it. Split the shape into triangles and add them, or take the six corner coordinates and run the shoelace formula, which handles any polygon regardless of shape.

Why do bees build hexagons?

Because among shapes that tile with no gaps, the hexagon encloses a given area with the least boundary. For 93.53 square units a hexagon needs a perimeter of 36 against a square’s 38.68, so less wax per cell.

Related lessons and calculators

• Area of an equilateral triangle, the single piece this whole shape is built from, six times over.
• Area formulas for every common shape, including the general regular polygon formula this is a special case of.
• Area of a rhombus, another shape where the easiest measurement to take is not the one in the usual formula.
• Area of a shape from coordinates, the shoelace route for hexagons that are not regular.
• Area of 2D shapes calculator, for running these without the arithmetic.
• the National Council of Teachers of Mathematics, for how regular polygons are introduced in the curriculum.