Quick answer: V = (3√3/2)a²h ≈ 2.598 a²h, where a is the side length and h the length of the prism. A hexagon of side 6 extended 10 long holds 935.31 cubic units. If you measured across the flats — as hex bar and nuts are specified — divide that by √3 first to get the side.
Which width did you measure?
This is where nearly every wrong hexagonal-prism answer starts. A hexagon does not have one width; it has two, and they differ by about 15%.
| Measurement | What it is | Relation to side a | Value |
|---|---|---|---|
| Side length a | One of the six edges | — | 6 |
| Across flats (AF) | Between opposite faces — what a spanner grips | a√3 | 10.3923 |
| Across corners (AC) | Between opposite points | 2a | 12 |
| Apothem | Centre to the middle of a face | a√3/2 | 5.1962 |
The ratio AC : AF is always 2/√3 ≈ 1.1547, whatever the size. Feed an across-corners figure into a formula expecting across-flats and the area comes out 33% too large, because area scales with the square of the width.
Engineering convention: across flats wins. Hexagonal bar stock, nuts, bolt heads and socket sizes are all specified across the flats. An "M10 nut, 17 mm" means 17 mm between opposite faces — the spanner size. If a supplier's datasheet says "hex 25 mm" with no qualifier, it means 25 mm AF. Across-corners is essentially only used when you need to know whether the part clears an opening.
Where the formula comes from
A hexagonal prism is the plainest kind of prism: cross-sectional area × length. All the interesting work is in the hexagon.
Draw lines from the centre to each of the six corners and the hexagon splits into six identical equilateral triangles, each with side a. That is a property unique to the regular hexagon — the distance from centre to corner happens to equal the side length. Each triangle has area (√3/4)a², so six of them give:
| Step | Expression | a = 6 |
|---|---|---|
| One equilateral triangle | (√3/4)a² | 15.5885 |
| Six of them = hexagon area | (3√3/2)a² ≈ 2.598a² | 93.5307 |
| × length h | (3√3/2)a²h | 935.3074 (h = 10) |
If you have the across-flats distance F instead, there is a tidier form that skips the conversion entirely: area = (√3/2)F². For F = 10 that gives 86.6025 — identical to working out a = 5.7735 first and using the standard formula. Both routes are built into the calculator.
Reference table
Computed at full precision and rounded only at the end.
| Side a | Across flats | Across corners | Hexagon area | Length h | Volume |
|---|---|---|---|---|---|
| 1 | 1.7321 | 2 | 2.5981 | 1 | 2.5981 |
| 2 | 3.4641 | 4 | 10.3923 | 5 | 51.9615 |
| 3 | 5.1962 | 6 | 23.3827 | 10 | 233.8269 |
| 4 | 6.9282 | 8 | 41.5692 | 6 | 249.4153 |
| 5 | 8.6603 | 10 | 64.9519 | 12 | 779.4229 |
| 6 | 10.3923 | 12 | 93.5307 | 10 | 935.3074 |
| 10 | 17.3205 | 20 | 259.8076 | 20 | 5196.1524 |
Read down the "across corners" column and the pattern is obvious — it is always exactly twice the side. That makes it the easiest of the three to work back from mentally: halve it and you have the side length.
Hex bar stock: volume to weight
The most common practical use of this calculation is costing or shipping hexagonal bar. Volume alone rarely settles it — you want the mass.
| AF size | Area | Volume per m | Steel (7.85 g/cm³) | Aluminium (2.70) | Brass (8.50) |
|---|---|---|---|---|---|
| 10 mm | 0.8660 cm² | 86.60 cm³ | 0.680 kg | 0.234 kg | 0.736 kg |
| 13 mm | 1.4636 cm² | 146.36 cm³ | 1.149 kg | 0.395 kg | 1.244 kg |
| 17 mm | 2.5028 cm² | 250.28 cm³ | 1.965 kg | 0.676 kg | 2.127 kg |
| 19 mm | 3.1264 cm² | 312.64 cm³ | 2.454 kg | 0.844 kg | 2.657 kg |
| 25 mm | 5.4127 cm² | 541.27 cm³ | 4.249 kg | 1.461 kg | 4.601 kg |
| 32 mm | 8.8681 cm² | 886.81 cm³ | 6.961 kg | 2.394 kg | 7.538 kg |
| 50 mm | 21.6506 cm² | 2165.06 cm³ | 16.996 kg | 5.846 kg | 18.403 kg |
Note that the area uses (√3/2)F², the across-flats form. A 25 mm AF bar has a cross-section of 5.4127 cm², so a metre of steel comes in at about 4.25 kg. Doubling the AF size quadruples the weight — area scales with the square, which is why 50 mm bar is four times the 25 mm figure, not twice.
Cutting hex from round bar wastes ~17%. A hexagon inscribed in a circle occupies (3√3/2)/π = 82.7% of it. If hex stock is machined down from round, roughly one part in six leaves as swarf — worth knowing when a supplier quotes hex at a premium over round of the same AF.
Hexagonal tanks, planters and containers
For anything holding liquid or soil, convert to capacity: 1 litre = 1000 cm³ exactly, and 1 m³ = 1000 litres.
| Step | Working | Result |
|---|---|---|
| Side length | measured | 30 cm |
| Hexagon area | 2.598 × 30² | 2338.27 cm² |
| Depth | measured | 40 cm |
| Full volume | 2338.27 × 40 | 93,530.74 cm³ |
| Capacity | ÷ 1000 | 93.53 litres |
| Realistic soil fill (37 cm) | 2338.27 × 37 ÷ 1000 | 86.52 litres |
That last row matters more than it looks. Filling to the brim is never what happens — planters need 3–5 cm of freeboard so watering does not wash soil over the edge, and tanks need headspace. Buying 93 litres of compost for an 86-litre hole means a spare bag; the full-volume figure is the wrong one to shop with.
Why hexagons, everywhere
Hexagonal prisms turn up far more often in nature and engineering than their awkward formula suggests, and there is a single reason: the hexagon is the most efficient shape that tiles a plane.
Of all shapes that can cover a surface with no gaps, the regular hexagon needs the least perimeter for a given area. For bees that means the least wax for the most honey. Conjectured for two millennia, it was finally proved by Thomas Hales in 1999 as the honeycomb conjecture.
| Shape | Perimeter | vs hexagon |
|---|---|---|
| Regular hexagon | 37.22 | — |
| Square | 40.00 | +7.5% |
| Equilateral triangle | 45.59 | +22.5% |
| Circle (does not tile) | 35.45 | −4.8% |
The circle beats the hexagon but leaves gaps, so it is disqualified. The hexagon is the best available answer, and that is why it shows up in honeycomb, basalt columns at the Giant's Causeway, graphite sheets, carbon-fibre core panels and nut geometry alike.
Common mistakes & pro tips
| Mistake | What happens | Fix |
|---|---|---|
| Across-corners entered as across-flats | Volume 33% too high | AC = 2a, AF = a√3; pick the matching mode above |
| Using the apothem as the side | Volume 25% too low | Apothem is a√3/2, not a |
| Treating the hexagon as a circle | Over-states by ~21% | A hexagon fills only 82.7% of its circumcircle |
| Multiplying area by 6 again | Six-fold error | The 2.598 coefficient already counts all six triangles |
| Using ⅓ in the formula | You used the pyramid formula | Prisms have no ⅓ — that belongs to pyramids and cones |
| Filling a planter to the calculated brim | Over-buys compost | Subtract 3–5 cm of freeboard |
Pro tip on measuring. Across-flats is far easier to measure accurately than the side length — callipers sit flat against two parallel faces with nothing to slip off. Measuring a single edge means finding two corners precisely, and corners on real parts are usually chamfered or rounded. Measure AF, let the calculator do the conversion.
How to use this calculator
- Choose which width you measured: side length, across flats (bar stock, nuts, spanner sizes) or across corners.
- Enter that measurement, then the length of the prism — the depth for a planter, the bar length for stock.
- Pick the output unit. The breakdown gives the hexagon area, all three widths, the surface area and the equivalent capacity.
- Optionally choose a material to get the weight, which is usually what you actually need for bar stock.
Frequently asked questions
What is the formula for the volume of a hexagonal prism?
Volume equals (3√3/2)a²h, or about 2.598 a²h. A regular hexagonal prism with a side of 6 and a length of 10 has a volume of 935.31 cubic units. The first part is just the area of the hexagonal end; the rest is standard prism logic — cross-sectional area times length.
What is the difference between across flats and across corners?
Across flats is the distance between two opposite parallel faces — the way a spanner grips a nut. Across corners is between two opposite points, and is always larger by 2/√3 ≈ 1.1547. For a side of 6, across flats is 10.39 and across corners is 12. Confusing them changes the volume by about 33%.
How do I find the volume of hex bar stock?
Hex bar is specified by its across-flats dimension — the same figure as the spanner size. Enter that, choose across-flats mode, then the bar length. A 25 mm AF bar one metre long is 541.27 cm³, which in steel weighs about 4.25 kg. Pick a material above and the calculator reports weight directly.
What is the area of a regular hexagon?
(3√3/2)a², about 2.598a². A hexagon of side 4 has an area of 41.57 square units. The formula comes from splitting the hexagon into six identical equilateral triangles, each of area (√3/4)a² — six of those give the coefficient.
How much water does a hexagonal tank or planter hold?
Get the volume in cm³ and divide by 1000, since 1 litre = 1000 cm³ exactly. A planter with a side of 30 cm and depth of 40 cm holds 93.53 litres brim-full. Fill only to the soil line — subtract 3–5 cm of depth for a realistic figure, which drops it to about 86.5 litres at 37 cm.
Why are honeycombs hexagonal?
Because the hexagon is the most efficient shape that tiles a plane with no gaps — it needs the least perimeter for a given area, so bees use the least wax for the most storage. Conjectured for two thousand years, it was proved by Thomas Hales in 1999 as the honeycomb conjecture. The same logic explains hexagonal patterns in graphite, basalt columns and carbon-fibre cores.
How much of a cylinder does a hexagon fill?
About 82.7%, since (3√3/2)/π = 0.827. That is why cutting hex bar from round stock wastes roughly 17% of the material, and why a hexagonal container inside a circular footprint gives up a similar share of capacity. Hexagons are efficient at tiling flat space, not at filling circles.
What is the surface area of a hexagonal prism?
Two hexagonal ends plus six rectangular sides: 3√3 a² + 6ah. For a side of 6 and length of 10, the ends contribute 187.06 and the sides 360, giving 547.06 square units. Leave the ends out if the prism is a length of bar joined at both ends.