One number fixes a regular hexagon
A regular hexagon is the six-sided figure with all sides equal and every interior angle 120°. That much symmetry leaves exactly one free parameter, so any single measurement — the side, the width across the flats, the point-to-point diagonal, the perimeter or the area — determines every other one. This calculator takes whichever you have and returns the rest.
The reason the hexagon turns up everywhere is that it tiles the plane with no gaps while using less edge length per unit area than a square or a triangle. Bees exploit that, and so do honeycomb cores, tile setters and mesh designers. It is also the standard cross-section for wrench-driven fasteners, because six flats give a driver six engagement positions every 60° instead of the four a square gives.
Split the hexagon from its centre to each vertex and you get six identical equilateral triangles of side s. That single observation gives you the area formula immediately: six times the area of an equilateral triangle. If you need that triangle on its own, the equilateral triangle calculator handles it, and for shapes with any other number of sides use the regular polygon area calculator.
Where (3√3 / 2)·s² comes from
Start with one of the six equilateral triangles. Its base is s and its height is the apothem, s·√3/2, because dropping a perpendicular from the apex splits it into two 30-60-90 triangles whose long leg is √3 times the short leg. So each triangle has area ½ · s · s√3/2 = (√3/4)s². Six of them give A = (3√3/2)·s² ≈ 2.598076·s². The 30-60-90 triangle calculator shows that side ratio in isolation.
Two hexagon widths matter in practice and they are not the same number. Across flats (F) is the perpendicular distance between two opposite parallel sides — twice the apothem, so F = √3·s ≈ 1.732s. That is what a caliper reads on hex bar and what a wrench size names. Across corners (C) is the long diagonal through the centre, which passes along two triangle sides, so C = 2s. The ratio C/F = 2/√3 ≈ 1.1547 is fixed: a point-to-point measurement is always about 15.5% larger than the wrench size.
Substituting s = F/√3 into the area formula collapses it to the form machinists prefer: A = (√3/2)·F² ≈ 0.866025·F². And because the perimeter is 6s and the apothem is F/2, the general polygon identity A = ½·P·a reproduces the same answer, which is a useful arithmetic check.
Worked example: the area of 1-inch hex bar
You have hex bar stock listed as 1 inch. That listing is the across-flats dimension, so F = 1.0000 in.
- Side length. s = F / √3 = 1.0000 / 1.732051 = 0.577350 in.
- Across corners. C = 2s = 2 × 0.577350 = 1.154701 in. So the bar will not pass through a 1.10 in round hole even though it is called 1 inch.
- Apothem. a = F/2 = 0.500000 in.
- Perimeter. P = 6s = 3.464102 in.
- Area, the long way. A = (3√3/2)s² = 2.598076 × 0.577350² = 2.598076 × 0.333333 = 0.866025 in².
- Area, the short way. A = (√3/2)F² = 0.866025 × 1.0000² = 0.866025 in². The two agree, as they must.
- Check with A = ½Pa. ½ × 3.464102 × 0.500000 = 0.866025 in². Three routes, one answer.
Now turn that into weight. Steel at 0.2836 lb/in³ gives 0.866025 × 0.2836 = 0.2456 lb per inch of length, so a 12-inch piece weighs about 2.95 lb. Compare it with 1-inch round bar, whose area is π/4 = 0.785398 in²: the hex carries about 10.3% more metal for the same nominal size, because 0.866025/0.785398 = 1.1027.
Reading the result: which width does your job need
Choose the dimension by the physical constraint. If something has to grip the hexagon — a wrench, a collet, a hex broach — the controlling number is across flats. If something has to clear it — a bore, a socket recess, a pocket milled to hold the part — the controlling number is across corners, and using the flats figure there is the classic way to make a part that will not fit.
For tile and paving, the area output is the one that drives quantity, but the across-flats figure drives layout. Hexagonal tiles pack in offset rows whose row-to-row spacing is 1.5s = 0.866F, not F, so a run of ten rows is shorter than ten tile widths. Divide your total surface by the tile area, then add cut waste on the perimeter; hexagons generate more edge cuts than squares because no cut edge along a straight wall is ever a full tile.
The apothem is the number to reach for when you are inscribing or circumscribing circles. A circle of radius equal to the apothem fits inside the hexagon touching all six flats; a circle of radius s passes through all six corners. The ratio of hexagon area to that circumscribed circle area is 3√3/(2π) = 0.8270, so a hexagon captures about 82.7% of its circumscribed circle — run the circle figure yourself in the circle area calculator if you are comparing stock shapes.
Common hex bar sizes: side, across corners and area
| Across flats F | Side s | Across corners C | Area (in²) |
|---|---|---|---|
| 0.2500 | 0.1443 | 0.2887 | 0.0541 |
| 0.3750 | 0.2165 | 0.4330 | 0.1218 |
| 0.5000 | 0.2887 | 0.5774 | 0.2165 |
| 0.6250 | 0.3608 | 0.7217 | 0.3383 |
| 0.7500 | 0.4330 | 0.8660 | 0.4871 |
| 1.0000 | 0.5774 | 1.1547 | 0.8660 |
| 1.2500 | 0.7217 | 1.4434 | 1.3532 |
| 1.5000 | 0.8660 | 1.7321 | 1.9486 |
Every row is s = F/√3, C = 2s and A = (√3/2)F². Multiply the area by material density to get weight per unit length.
Mistakes that produce the wrong hexagon
- Treating the wrench size as the diagonal. Across corners is 2/√3 = 1.1547 times across flats. A pocket cut to the flats dimension will not accept the part.
- Using s = F/2. That is the apothem, not the side. The side is F/√3, about 15.5% larger than half the width.
- Applying these formulas to an irregular hexagon. Every result here assumes six equal sides and six 120° angles. For an irregular six-sided outline, take coordinates and use the shoelace formula instead.
- Ignoring the corner radius on real stock. Extruded and drawn hex bar carries a small radius at each corner, so its true area is slightly below the geometric figure and its across-corners measurement slightly below 1.1547F.
- Mixing units inside one calculation. This tool is unit-free by design: whatever unit you type in is the unit that comes back, and areas come back squared in it.
Related shapes and when to switch tools
The hexagon is the n = 6 case of the general regular polygon, whose area is A = ¼·n·s²·cot(π/n). Put n = 6 in that and cot(30°) = √3 returns (3√3/2)s² exactly. If you are comparing a hexagon against a pentagon or an octagon — for a bolt head, a gazebo deck or a planter — the regular polygon area calculator handles any n, and the polygon interior angle calculator gives the 120° figure and its relatives.
Three regular polygons tile the plane on their own: the triangle, the square and the hexagon. Among those three the hexagon has the shortest total edge length for a given enclosed area, which is why honeycomb structures and cell-network diagrams use it. That property is the discrete cousin of the isoperimetric result that a circle encloses the most area per unit perimeter, and the hexagon gets to 82.7% of the way there while still tiling.
For layout work you will normally pair this with two other tools. The triangle area calculator covers the six wedges individually when you are cutting a hexagon out of triangular offcuts, and the circumference calculator gives the inscribed and circumscribed circles when you are marking a hexagon out with a compass — the classic construction steps a compass set to the radius six times around the circle, which works precisely because s equals the circumradius.