What this calculator does
A hexagonal pyramid is a solid with a regular six-sided base and six triangular faces meeting at a single apex directly above the centre of the base. This hexagonal pyramid calculator finds its volume and surface area from just two measurements: the length of one base edge and the vertical height from the base to the apex.
The working leans on one property of a regular hexagon that makes the geometry manageable: its circumradius (centre to corner) is exactly equal to its side length, and its apothem (centre to the middle of a side) is (√3/2) times the side length. Those two distances, combined with the pyramid's height, give the slant height and lateral edge by simple right-triangle relationships, without any extra measurements needed.
The formula
The base area of a regular hexagon with edge a is (3√3/2)a². The apothem, the perpendicular distance from the centre to the middle of a side, is (√3/2)a. The slant height, from the apex down to the midpoint of a base edge, forms a right triangle with the pyramid's height and the apothem, so it equals the square root of height squared plus apothem squared. Volume follows the standard pyramid rule, one third of base area times height, and the lateral surface area is three times the base edge times the slant height (six triangular faces, each with area ½ × edge × slant height).
| Term | Meaning |
|---|---|
| a | The length of one edge of the regular hexagonal base. |
| h | The pyramid's vertical height, from the centre of the base straight up to the apex. |
| Apothem | The perpendicular distance from the centre of the hexagon to the midpoint of one of its sides, equal to (√3/2)a. |
| Slant height | The distance from the apex down to the midpoint of a base edge, measured along the face. |
| Lateral edge | The distance from the apex down to one of the base's six corners. |
The inputs explained
| Field | What to enter |
|---|---|
| Base edge length | The length of one side of the hexagonal base. All six sides are equal, since the base is a regular hexagon. |
| Height | The pyramid's vertical height, measured straight up from the centre of the base to the apex, not along a sloped face. |
When to use it
Volume of a hexagonal container or crystal model
Hexagonal pyramid shapes turn up in packaging, architectural models and crystal geometry. Volume from a base edge and height gives the capacity or material volume directly.
Surface area for material or coating estimates
Total surface area, base plus the six triangular faces, is what determines how much material covers the outside of the shape, useful for anything from papercraft nets to surface coatings.
Comparing against a square pyramid of similar size
The same base-edge-and-height approach used for a square pyramid applies here, just with the hexagon's different base area and apothem formulas, making it straightforward to compare how much more volume a hexagonal base adds over a square one of a similar span.
Worked examples
Every figure in the tables below is produced by this page’s own calculator at build time, so the numbers and the tool always agree. Select any row to load that scenario.
How do volume and surface area change with a taller hexagonal pyramid?
Volume and surface area for a hexagonal pyramid with a 6-unit base edge, as height increases.
| Height | Total surface area | Slant height (apex to mid-edge) |
|---|---|---|
| 4 | 211.565 | 6.557 |
| 6 | 236.401 | 7.937 |
| 8 | 265.240 | 9.539 |
| 10 | 296.380 | 11.269 |
| 15 | 379.272 | 15.875 |
| 20 | 465.482 | 20.664 |
Questions
Why is the circumradius of a regular hexagon equal to its side length?
A regular hexagon can be split into six equilateral triangles meeting at the centre. Each triangle has all three sides equal, including the two radii, which means the radius from the centre to a corner is exactly the same length as a side.
What is the difference between the slant height and the lateral edge?
The slant height runs from the apex to the middle of a base edge, along the flat face. The lateral edge runs from the apex to a corner of the base. They are different lengths because the apothem (used for slant height) is shorter than the circumradius (used for lateral edge).
Does this work for an irregular hexagonal base?
No, the formulas here assume a regular hexagon, with all six sides and angles equal. An irregular base needs its own base area calculated separately before the same pyramid volume rule (one third base area times height) can be applied.
How does this compare to a square pyramid of the same base edge and height?
A regular hexagon has a larger area than a square with the same edge length, so a hexagonal pyramid holds more volume than a square pyramid built from the same base edge and height, as shown by comparing this calculator against the square pyramid volume calculator.
For the four-sided equivalent, see the square pyramid volume calculator and the square pyramid surface area calculator.