Regular hexagon calculator
Tell the tool what you know — the side length, the long or short diagonal, or the area — and it derives every other property of a regular hexagon: area, perimeter, both diagonals and the apothem. It is useful for tiling, nut-and-bolt sizing, board-game grids, and geometry homework.
How it works
The calculator first solves for the side length from whatever you entered, then computes the rest. A regular hexagon is six equilateral triangles meeting at the centre, which gives these exact relationships (where s is the side):
- Area = (3√3 / 2) · s² ≈ 2.598 · s²
- Perimeter = 6 · s
- Long diagonal (vertex to vertex) = 2 · s
- Short diagonal (flat to flat, the width across flats) = s · √3 ≈ 1.732 · s
- Apothem (centre to side midpoint) = (√3 / 2) · s ≈ 0.866 · s
If you start from the area, it reverses the area formula: s = √(2·area / (3√3)).
Example
For a hexagon with side = 10:
| Property | Formula | Value |
|---|---|---|
| Area | 2.598 × 10² | 259.81 |
| Perimeter | 6 × 10 | 60 |
| Long diagonal | 2 × 10 | 20 |
| Short diagonal | 10 × √3 | 17.32 |
| Apothem | 0.866 × 10 | 8.66 |
All calculations stay in your browser; nothing is uploaded.
Why hexagons appear so often in engineering and nature
The regular hexagon is geometrically special: it is one of only three regular polygons that tile a flat surface without gaps (the others are equilateral triangles and squares). Among those three, hexagons pack the most area for the least perimeter — a fact that honeybees exploit by building honeycomb cells. This efficiency makes the hexagon a recurring shape in engineering and design:
- Nuts and bolts use hexagonal heads because a 6-point wrench gives more engagement angles than a square or octagonal head. When engineers specify “width across flats” for a fastener, that measurement is the short diagonal of a regular hexagon.
- Board games and strategy simulations use hex grids because every cell is equidistant from all six neighbours, which eliminates the diagonal-vs-orthogonal distance problem that square grids introduce.
- Structural panels — in aircraft and satellites, hexagonal honeycomb cores between thin sheets of aluminium or carbon fibre provide high stiffness at low weight precisely because of the hexagon’s perimeter efficiency.
- Graphene and carbon nanotubes have a hexagonal lattice at the atomic level, which is part of what gives them their extraordinary strength.
Working backwards from measurements you have
If you are sizing a hexagonal space (tiling a floor, cutting a gasket, specifying a recess), you usually start from the flat-to-flat dimension, which is the short diagonal. Enter that value in the short-diagonal field and the tool derives the side, area, and apothem for you. If you know the corner-to-corner span instead, enter the long diagonal. The area-to-side reversal is particularly useful for territory-planning problems: if you know a hexagonal cell must cover a given area, the calculator tells you how long each side must be.
Relationship between the two diagonals and the side
There is a clean ratio worth remembering: the long diagonal is always exactly twice the side, and the short diagonal is the side times √3 (approximately 1.732). So if you know any one of the three, you know all three without a calculator. The apothem is half the short diagonal — this is useful for fitting circles inside a hexagon or checking if a round component clears the flat walls of a hexagonal hole.