Hexagon Calculator
Enter any one measurement of a regular hexagon and every other dimension is calculated instantly. Solve from the side, area, perimeter, long or short diagonal, apothem, or circumradius. Switch between metric and imperial units, then check the step-by-step working below.
Formula
Worked example
For a = 6 cm: area = (3√3/2) × 36 = 54√3 ≈ 93.53 cm². Perimeter = 36 cm. Long diagonal = 12 cm. Short diagonal = 6√3 ≈ 10.39 cm. Apothem = 3√3 ≈ 5.196 cm. Circumradius = 6 cm. Reverse: if the across-flats width is 10.39 cm, the side is 10.39 / √3 ≈ 6 cm.
Why every hexagon measurement comes from one number
A regular hexagon has six equal sides and six interior angles of exactly 120 degrees. That perfect symmetry means knowing any single measurement fixes the entire shape. The calculator exploits the six exact algebraic relationships below to solve forward (from side to all others) or backward (from any measured quantity to the side, then to everything else). The area formula, A = (3√3/2)a², comes from dividing the hexagon into six congruent equilateral triangles, each with area (√3/4)a². The perimeter is simply 6a. The long diagonal, which goes from one corner straight through the centre to the opposite corner, equals exactly 2a, making the circumradius equal to a. The short diagonal, skipping one corner, equals a√3; this is also called the across-flats distance and is the figure stamped on hex bolt heads and hex nuts.
Solving from any measurement (reverse mode)
Most online hexagon tools only work one way: enter the side, get everything else. This calculator also reverses any of the six formulas: - From area: a = √(2A / (3√3)) - From perimeter: a = P / 6 - From long diagonal: a = d / 2 - From short diagonal: a = s / √3 - From apothem: a = 2r / √3 - From circumradius: a = R (they are identical for a regular hexagon) Practically this is useful whenever you measure an existing hexagon. A machinist reads the hex across the flats (the short diagonal) on a bolt and needs the nominal side. A tile installer measures a hex tile tip-to-tip (the long diagonal). An engineer reads the inradius off a CAD drawing. Switch the "Solve from" selector to whichever measurement you have and the rest follows automatically.
Unit switching and tiling cost estimate
Toggle between metric (centimetres by default) and imperial (inches) at the top of the calculator to keep all inputs and outputs in the same system. The labels update throughout so you always know which unit you are working in. The optional tiling cost estimate answers a practical question: if these tiles are hexagonal, how many do I need to cover a given floor or wall area, and what will they cost? Enter the area to tile, the price per tile, and a waste allowance (typically 10% for cuts and breakages around edges). The calculator divides the total area by the area of one tile, applies the waste factor, and rounds up to whole tiles. Make sure the tile side length and the floor area are in consistent units, for example both in centimetres or both in inches.
Hexagons in nature and engineering
Regular hexagons tile a flat surface with no gaps and no overlaps, sharing that property with only the equilateral triangle and the square. Among those three, the hexagon encloses the most area for a given perimeter, which is why honeybees build hexagonal cells: they minimise wax usage per unit of honey stored. The same tessellation efficiency drives the use of hexagons in heat exchangers, carbon nanotube lattices (graphene), and the reflecting mirror segments of space telescopes such as the James Webb Space Telescope, whose primary mirror is made of 18 hexagonal gold-coated beryllium segments.
Regular hexagon: formulas and multipliers
| Quantity | Forward (from a) | Reverse (to find a) | Value for a = 1 |
|---|---|---|---|
| Area (A) | (3√3/2) × a² | a = √(2A / (3√3)) | 2.5981 |
| Perimeter (P) | 6 × a | a = P / 6 | 6.0000 |
| Long diagonal (d) | 2 × a | a = d / 2 | 2.0000 |
| Short diagonal (s) | √3 × a | a = s / √3 | 1.7321 |
| Apothem (r) | (√3/2) × a | a = 2r / √3 | 0.8660 |
| Circumradius (R) | a | a = R | 1.0000 |
All quantities expressed as multiples of the side a (area scales with a²). Reverse formulas let you recover a from any measured quantity.
Frequently asked questions
How do I find the area of a regular hexagon?
For a regular hexagon with side length a, the area is (3√3/2) × a², approximately 2.598 × a². Dividing the hexagon into six equilateral triangles of side a, each with area (√3/4)a², and multiplying by 6 gives the same result. If you know the apothem r instead, use A = r × P / 2 = r × 6a / 2 = 3ar.
What is the difference between the long and short diagonal?
The long diagonal (also called the main diagonal or diameter) connects two opposite vertices through the centre and measures exactly 2a, where a is the side length. The short diagonal connects two vertices with one vertex between them and measures a√3 ≈ 1.732a. A regular hexagon has exactly three long diagonals (all equal) and six short diagonals (all equal). The short diagonal is also the across-flats width.
How do I find the side length from the across-flats width?
The across-flats width of a regular hexagon is its short diagonal, which equals a√3. To recover the side length, divide the across-flats measurement by √3 (approximately 1.73205). For example, a hex bolt measuring 19 mm across the flats has a nominal side of 19 / 1.73205 ≈ 10.97 mm. Use the "Short diagonal" option in this calculator to do this automatically.
Is the circumradius always equal to the side length?
Yes, for a regular hexagon only. The circumradius R (distance from the centre to any vertex) equals the side length a exactly. This is why a pair of compasses set to the side length and swept around a circle of that radius marks out six perfect vertices. No other regular polygon shares this property.
How many tiles do I need to cover a floor with hexagonal tiles?
Divide the total floor area by the area of one tile to get the minimum count, then multiply by a waste factor (typically 1.10 for 10% cutting waste around edges and corners). Always round up to a whole number of tiles. The tiling cost section in this calculator does this for you: enter the floor area, price per tile, and waste allowance.
What is the apothem of a hexagon?
The apothem (also called the inradius) is the perpendicular distance from the centre of the hexagon to the midpoint of any edge. For a regular hexagon with side a, the apothem is (√3/2)a ≈ 0.866a. It is also the radius of the largest circle that fits inside the hexagon (the inscribed circle). The apothem equals half the short diagonal.