The hexagon calculator above solves a regular six-sided figure from any one of five different measurements. That flexibility matters more here than for most shapes, because hexagons are usually specified by a width rather than an edge — a hex nut is sold by its across-flats size, a hex tile by its point-to-point width, and neither of those is the side length the textbook formula wants.
Arb Digital publishes this alongside our other free geometry tools so that the conversion between those specifications is done for you rather than half-remembered. Enter whichever number is on your drawing or product listing, and the calculator fills in the rest along with the working.
What This Hexagon Calculator Does
It returns the area, perimeter, apothem, circumradius, across-flats width and across-corners width of a regular hexagon — one with six equal sides and six equal 120° angles. Enter a side length of 10 cm and you get an area of about 259.81 cm², a perimeter of 60 cm, an apothem of 8.66 cm, an across-flats width of 17.32 cm and an across-corners width of exactly 20 cm.
The quantity field multiplies the area across several hexagons, which is how you get a tiling or panel total without a second pass. Note that the calculator covers regular hexagons only. An irregular six-sided outline needs corner coordinates instead, which our polygon calculator handles with the shoelace method.
How to Use It
- Identify which measurement you actually have. Look at the drawing or spec sheet: "A/F" or "across flats" means the flat-to-flat width, "A/C" means point-to-point.
- Select that measurement type from the dropdown so the calculator converts correctly rather than assuming you meant the side.
- Enter the value and choose a unit. All results come back in that unit, with the area squared.
- Set a quantity if you are covering an area with repeated hexagons.
- Check the working line — it restates the conversion and the area calculation with your own numbers so you can verify it.
The Formula: How Hexagon Area Is Calculated
The area of a regular hexagon with side s is A = (3√3 / 2) s², which is approximately 2.598076 × s². For s = 10 cm, A = 2.598076 × 100 = 259.81 cm². The constant comes straight from cutting the hexagon into six equilateral triangles of side s, each with area (√3/4)s². Six of those is (6√3/4)s² = (3√3/2)s², which is where the awkward-looking number comes from.
The general polygon formula gives the same answer by a different route: A = ½ × perimeter × apothem. With a perimeter of 60 cm and an apothem of 8.660 cm, that is ½ × 60 × 8.660 = 259.81 cm² — matching to the decimal place. The apothem itself is s√3/2, since it is the height of one of those equilateral triangles. MathWorld's hexagon entry lists the full set of relations.
Across Flats vs Across Corners: The Spec Sheet Trap
This is the single most useful thing on the page for anyone working with real parts. A regular hexagon has two different widths depending on which way you measure it, and they differ by about 15%. Across corners is twice the side length: A/C = 2s. Across flats is twice the apothem: A/F = s√3 ≈ 1.7321s. The ratio between them is 2/√3 ≈ 1.1547.
Fasteners are almost always specified across flats, because that is the dimension a spanner or socket grips. An M10 hex nut with a 17 mm across-flats size has a side length of 17 ÷ 1.7321 ≈ 9.815 mm and an across-corners size of about 19.63 mm — which is the clearance you need in a recess, not 17 mm. Tiles, paving and mesh, by contrast, are often quoted across corners or by side length. Reading one figure as the other produces a part that does not fit, or an order roughly a third short on area, since area scales with the square of the linear error.
Why Hexagons Tile So Well
Only three regular polygons tile a plane with no gaps: the triangle, the square and the hexagon. The hexagon is the efficient one. Among all ways of dividing a plane into equal-area cells, the regular hexagonal grid has the shortest total boundary — a result long observed in honeycombs and finally proved by Thomas Hales in 1999, described at MathWorld's honeycomb conjecture page.
The practical consequence is that hexagonal cells enclose the most area for the least dividing material. That is why bees build hexagonal comb, why lightweight structural cores are hexagonal, and why hex mesh and hex paving are used where material cost per unit of covered area matters. It is also why hexagonal tiles produce less grout line per square metre than the same area of small square tiles, which affects both cost and cleaning.
Angles, Diagonals and Symmetry
Every interior angle of a regular hexagon is 120° and every exterior angle is 60°. The interior angles sum to (6 − 2) × 180° = 720°. Three hexagons meeting at a point make exactly 360°, which is precisely why the tiling closes with no gap — the same test fails for a pentagon, whose 108° angles leave a 36° shortfall.
A hexagon has nine diagonals in two lengths. The three long ones pass through the centre and measure 2s; they are the across-corners dimension. The six short ones skip one vertex and measure s√3, the same as the across-flats width. The hexagon has six lines of symmetry and six-fold rotational symmetry, and it is the only regular polygon whose side length equals its circumradius — which is the reason you can construct one with nothing but a compass set to a fixed radius, stepping around the circle six times.
Hexagons in Layout and Tiling Work
Laying out a hexagonal grid uses two spacings, not one. In a flat-top arrangement, the horizontal spacing between column centres is 1.5s and the vertical spacing between row centres is s√3, with alternate columns offset by half a row. Getting these two numbers right is the whole job; guessing at a single uniform spacing produces a grid that drifts out of alignment across a large floor.
For a real tiling estimate, calculate the area of one hexagon here, multiply by the tile count, and then take that figure to a materials tool. Our tile calculator handles grout gaps and waste allowance, and the flooring calculator covers floor coverings more broadly. Hexagonal tiles typically need a slightly higher waste allowance than square ones because edge cuts along a straight wall always waste part of a tile.
Hexagonal Grids in Mapping and Design
Hexagonal grids have quietly become the default for a whole class of data work, and the reason is a property of the shape rather than a fashion. In a square grid, a cell has eight neighbours at two different distances: four edge neighbours and four diagonal ones that are √2 times further away. That inconsistency distorts anything that depends on distance — heat maps, spread models, movement costs. In a hexagonal grid every cell has exactly six neighbours, all at the same centre-to-centre distance. There is no diagonal case to special-case.
This is why hex binning is used for dense scatter plots, why geospatial indexing systems build on hexagonal cells, and why strategy games have used hex maps since long before either. It is also why hexagons show up in structural panelling and acoustic tiles: equal spacing in six directions distributes load and diffusion more evenly than a square arrangement of the same cell size.
If you are laying out such a grid, the area figure from this calculator is your cell area, and the across-flats and across-corners readings give the two spacings you need. For a grid covering a known region, divide the region area by one cell's area to get a first estimate of the cell count, then add for the partial cells that will be cut off at the boundary.
Comparing Hexagons With Other Regular Polygons
For a fixed side length, area rises steeply with the number of sides: an equilateral triangle of side 10 has an area of about 43.3, a square 100, a pentagon about 172.05, a hexagon 259.81 and an octagon about 482.84. For a fixed perimeter the picture reverses in interest — more sides always enclose more area, approaching a circle as the limit. A hexagon of perimeter 60 encloses 259.81 square units; a circle of the same circumference encloses about 286.48, roughly 10% more. That gap is the price of having straight edges.
If you need a different polygon, we have dedicated pages for the pentagon calculator and the octagon calculator, and the circle calculator for the limiting round case. To convert a finished area between metric and imperial units, use the area converter rather than converting the side length and recalculating.
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Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Entering an across-flats width as the side length — that overstates the area by a factor of three, because across flats is roughly 1.73 side lengths.
- Using the apothem where the circumradius belongs — they differ by a factor of √3/2, about 13.4%, and both are measured from the centre.
- Assuming an irregular hexagon uses this formula — six sides is not enough. All six must be equal and all six angles 120°.
- Rounding √3 to 1.7 early — small errors in a linear dimension are squared in the area, so keep full precision until the final step.
- Ordering tiles equal to the bare calculated area — hexagonal edge cuts waste more material than square tiles do, so add a proper allowance.
Related Free Tools From Arb Digital
Handle any number of sides with the polygon calculator, measure boundaries with the perimeter calculator, work in straight rectangles with the rectangle area calculator, convert linear dimensions with the length converter, and check angle units with the angle converter. The full list lives in our free online tools hub.
Frequently Asked Questions
Area equals three times the square root of three, divided by two, multiplied by the side length squared — roughly 2.598076 × s². A hexagon with 10 cm sides has an area of about 259.81 cm².
Across flats is the width from one flat edge to the opposite flat edge, equal to the side length times the square root of three. Across corners is the point-to-point width, equal to twice the side length. Across corners is about 15.5% larger.
It is the perpendicular distance from the centre to the middle of any side, equal to the side length times the square root of three, divided by two. Half the across-flats width is the same thing.
Because a regular hexagon divides into six equilateral triangles meeting at the centre. Each triangle has the circumradius as two of its sides and the hexagon edge as the third, so all three are equal. That property is what lets you construct a hexagon with a fixed compass setting.
Each interior angle is 120° and each exterior angle is 60°. The six interior angles sum to 720°.
A hexagonal grid divides a plane into equal-area cells using the least total wall length of any possible arrangement. That result, known as the honeycomb conjecture, was proved in 1999. Less wall material for the same stored volume is a direct saving in wax.
No. It assumes six equal sides and six equal angles. For an irregular six-sided shape, enter the corner coordinates into our polygon calculator, which uses the shoelace formula and works for any outline.
