What is the formula for the area of a regular hexagon?

Area = (3 times the square root of 3, divided by 2) times the side squared. Written compactly: A = (3*sqrt(3)/2) * s^2, which is approximately 2.598 times s squared. The formula comes from dividing the hexagon into six identical equilateral triangles, each with area (sqrt(3)/4) * s^2, giving a total of 6 * (sqrt(3)/4) * s^2 = (3*sqrt(3)/2) * s^2.

How do I find the area if I only know the apothem?

First recover the side: s = 2 * apothem / sqrt(3). Then use the area formula: A = (3*sqrt(3)/2) * s^2. Alternatively, area also equals the perimeter times the apothem divided by 2, which is the general polygon area rule: A = (1/2) * P * a = (1/2) * 6s * a = 3 * s * a.

What is the difference between the long diagonal, the short diagonal, and the apothem?

The long diagonal connects two opposite vertices and equals twice the side length (d = 2s). The short diagonal connects the midpoints of two opposite edges (flat-to-flat width) and equals s times sqrt(3) (approximately 1.732 times s). The apothem is the perpendicular from the centre to the middle of any side and equals (sqrt(3)/2) times s, exactly half the short diagonal. These three measurements are all you need to reconstruct every dimension of a regular hexagon.

Can I work backwards from the area to find the side length?

Yes. Rearrange A = (3*sqrt(3)/2) * s^2 to get s = sqrt(2A / (3*sqrt(3))). This tool does that automatically when you select Area in the related hexagon calculator. Here you can enter the side, apothem, long diagonal, short diagonal, or perimeter, and the area is always the primary output.

Why does a regular hexagon have such a clean area formula?

A regular hexagon tiles perfectly because it is exactly six equilateral triangles joined at a central point. Because all six triangles share the same side length s, their combined area is six times (sqrt(3)/4)*s^2, which simplifies to (3*sqrt(3)/2)*s^2. No rounding, no approximation — the formula is exact for any value of s.

Is there a quick rule of thumb for the hexagon area?

Yes: multiply the side squared by 2.598. So a hexagon with a 10 cm side has an area of about 2.598 times 100 = 259.8 cm squared. For mental arithmetic, 2.6 times s squared is accurate to within 0.1%.

Hexagon Area Calculator

A hexagon area calculator that accepts any measurement you happen to have — side length, apothem, long diagonal, short diagonal, or perimeter — and returns the exact area of a regular hexagon instantly. It also shows an SVG diagram with the key dimensions labelled, a step-by-step working panel, and copy buttons on every result. Everything runs in your browser; no data is ever sent to a server.

What is a regular hexagon?

A regular hexagon is a six-sided polygon where every side is the same length and every interior angle is 120 degrees. It has a special place in geometry because it tiles a plane perfectly with no gaps — honeycombs, floor tiles, and circuit-board pads all exploit this property. It is also the most efficient shape that can be cut from a circle, since a regular hexagon inscribed in a circle uses about 82.7% of the circle’s area.

The shape has three pairs of parallel sides. The distance between one pair of parallel sides is the short diagonal (also called the width across flats, or the wrench size for a hex nut). The distance between two opposite vertices is the long diagonal. The perpendicular from the centre to the midpoint of any side is the apothem (inradius).

How the area formula is derived

A regular hexagon can be cut into six congruent equilateral triangles, each with side length s. The area of one equilateral triangle is (sqrt(3)/4) * s^2. Multiply by six:

A = 6 * (sqrt(3)/4) * s^2
  = (6*sqrt(3)/4) * s^2
  = (3*sqrt(3)/2) * s^2
  ≈ 2.598076 * s^2

The calculator uses this exact formula with full floating-point precision. If you start from a measurement other than the side length, the tool first solves for s using the relevant inverse formula, then applies the area equation.

Solve-for-variable shortcuts

What you know	Formula to get side s
Side length s	Direct
Apothem a	s = 2a / sqrt(3)
Long diagonal d	s = d / 2
Short diagonal w	s = w / sqrt(3)
Perimeter P	s = P / 6

Once s is known, every other property follows:

Area = (3*sqrt(3)/2) * s^2
Apothem = (sqrt(3)/2) * s
Long diagonal = 2 * s
Short diagonal = s * sqrt(3)
Perimeter = 6 * s

Worked example

A hexagonal floor tile has a short diagonal (flat-to-flat width) of 20 cm. What is its area?

Find the side: s = 20 / sqrt(3) = 20 / 1.73205… = 11.547 cm
Apply the area formula: A = (3*sqrt(3)/2) * 11.547^2 = 2.598 * 133.33 = 346.41 cm^2
The apothem is (sqrt(3)/2) * 11.547 = 10.0 cm (half the short diagonal, as expected).

You can verify this instantly: enter 20 in the calculator with “Short diagonal” selected and unit cm — the area should read 346.41 cm^2 at six decimal places.

Known input	Value	Resulting area
Side	10 cm	259.808 cm^2
Apothem	10 cm	346.410 cm^2
Long diagonal	20 cm	259.808 cm^2
Short diagonal	20 cm	346.410 cm^2
Perimeter	60 cm	259.808 cm^2

Practical applications

Hex nuts and bolts: the short diagonal is the wrench/spanner size; the area formula tells you the cross-sectional material area of the fastener head.
Tiling and paving: hexagonal tiles are specified by the flat-to-flat width; the area lets you calculate how many tiles cover a floor.
Board-game grids: wargames and strategy games use hexagonal grids where each hex represents a real-world area — knowing the tile area at scale gives the ground coverage.
Honeycomb structures: beehive cells and lightweight aerospace panels use regular hexagons because they maximise enclosed area for a given perimeter length.
PCB design: hexagonal copper pads pack more tightly than circular ones; the area formula determines copper coverage percentage.