How this instrument works
A hexagonal pyramid has a regular six-sided base and six triangular faces meeting at a single apex above it. Its volume relies on two separate facts stacked together: first, a regular hexagon's own area formula, (3√3 ⁄ 2)s², where s is the length of one edge; second, the universal pyramid rule that volume equals one third of the base area times the height. Combined, V = (3√3 ⁄ 2)s² × height ⁄ 3.
The hexagon area formula itself comes from splitting the regular hexagon into six identical equilateral triangles that meet at its center — each with area (√3 ⁄ 4)s², and six of them together give (6√3 ⁄ 4)s², which simplifies to (3√3 ⁄ 2)s². That same six-triangle decomposition is why a regular hexagon is one of the few polygons whose area formula reduces this cleanly to a single side length, alongside the equilateral triangle and the square.
This calculator assumes the pyramid's base is a REGULAR hexagon, with every side and every interior angle equal. An irregular hexagonal base — one with unequal sides — would need its own area computed by another method first (splitting it into triangles individually, for instance) before the same one-third pyramid rule could be applied to find the volume.
- Enter the regular hexagonal base's edge length into the Base edge length field.
- Enter the pyramid's height, the perpendicular distance from the apex down to the base's plane, into the Height field.
- Read Volume: the sheet computes the hexagon's own area first, then applies the one-third pyramid rule.
Worked example — a hexagon of edge 4, height 6
A hexagonal pyramid has a regular hexagonal base with a 4-unit edge and a height of 6 units. The base's own area is (3√3 ⁄ 2) × 4² ≈ 41.57 square units, and the volume is that area carried through the one-third pyramid rule: V ≈ 41.57 × 6 ⁄ 3 ≈ 83.14 cubic units.
Compare a smaller base with a 2-unit edge and a taller height of 9 units: the base area shrinks to (3√3 ⁄ 2) × 2² ≈ 10.39 square units, but the volume, V ≈ 10.39 × 9 ⁄ 3 ≈ 31.18 cubic units, is reached through very different proportions — a narrower base compensated by a much greater height.
Questions
What is the formula for the volume of a hexagonal pyramid?
V = (3√3 ⁄ 2)s² × height ⁄ 3, where s is the regular hexagonal base's edge length and height is the perpendicular distance from the apex to the base. The (3√3 ⁄ 2)s² portion is simply the hexagon's own area formula, carried through the standard one-third pyramid rule.
Does this formula work for an irregular hexagonal base?
No — it assumes a REGULAR hexagon, with all six sides and all six interior angles equal, since that regularity is exactly what lets the base area collapse to a single-edge-length formula. An irregular hexagonal base needs its area found separately, by another method, before the one-third pyramid rule can be applied.
Where does the 3√3 ⁄ 2 factor come from?
From splitting a regular hexagon into six identical equilateral triangles meeting at its center. Each triangle has area (√3 ⁄ 4)s², and six of them combine to (6√3 ⁄ 4)s², which simplifies to (3√3 ⁄ 2)s² — the regular hexagon's own area formula.
Why is there a division by 3 in a pyramid's volume formula?
Because exactly three pyramids sharing the same base and height can be assembled, with no gaps or overlaps, into a prism of that same base and height — a general geometric fact true for any base shape, including a hexagon, and provable through coordinate integration.
What if the base edge length is zero?
The hexagonal base collapses to a single point with zero area, so the pyramid's volume is zero regardless of the height — a fully degenerate, collapsed pyramid.