Hexagonal Pyramid Volume Calculator

A hexagonal pyramid has a regular hexagonal base and six triangular faces meeting at a single apex.

V = (√3 / 2) × a² × h ≈ 0.866 × a² × h

Where a is the hexagonal base edge and h is the perpendicular height from base center to apex.

The √3/2 factor comes from one-third of the hexagonal area (3√3/2 × a²): V = (1/3) × base × height = (1/3) × (3√3/2 × a²) × h = (√3/2) × a² × h.

Worked example: a hexagonal garden pavilion A six-sided garden pavilion: hexagonal floor with edge a = 2 m, roof apex 3 m above the floor (h = 3 m). V = 0.866 × 4 × 3 ≈ 10.39 m³ of enclosed air volume.

That is roughly 367 cubic feet, small enough for a portable heater to hold temperature without much effort.

Worked example: a pyramidal roof over a hexagonal hall A hexagonal roof with a = 8 m base edges and h = 5 m of rise: V = 0.866 × 64 × 5 ≈ 277 m³ of attic space.

Insulation, though, is bought by the square metre, not the cubic. The apothem here is 6.93 m, so the slant height is √(25 + 48) = 8.54 m and the six sloping faces come to 3 × 8 × 8.54 = 205 m². That, not the 277 m³, is the number the insulation and membrane quotes are built on. Volume tells you what the heating has to warm; area tells you what the job costs.

Where hexagonal pyramids appear:

• Garden pavilions and gazebos. Six-sided ones are common in formal gardens, more interesting than a square and easier to frame than an octagon.
• Pyramidal roof forms. Hexagonal towers in castles, observatories, and Victorian architecture often have hexagonal-pyramid roofs.
• Honeycomb-style structures. Some experimental architecture uses hexagonal pyramidal modules.
• Decorative crystal forms. Some crystals (apatite, beryl) develop hexagonal pyramidal terminations.
• Six-sided pencil tips (after sharpening). The exposed wood-and-lead point is a hexagonal pyramid.
• Faceted gemstones. Many gem cuts include hexagonal pyramid sections.

Comparing to other pyramids:

For the same base edge a and height h:

• Triangular pyramid (equilateral base): V = (√3/12) × a² × h ≈ 0.144 × a² × h
• Square pyramid: V = (1/3) × a² × h ≈ 0.333 × a² × h
• Pentagonal pyramid: V ≈ 0.573 × a² × h
• Hexagonal pyramid: V ≈ 0.866 × a² × h
• Octagonal pyramid: V ≈ 1.609 × a² × h

More sides, more volume, for a fixed edge length. That follows straight from the base: a hexagon of side a simply covers far more ground than a triangle of side a.

Hold the FOOTPRINT fixed instead, by inscribing every base in the same circle, and the comparison tightens right up. Each polygon closes in on that circle as sides are added, so the pyramids converge on the same cone.

A claim to be careful with

You will read that the hexagonal pyramid is the most volume-efficient shape for a given surface area. It is not, and this page’s own table is the proof: the octagonal pyramid beats it, and every step up in side count beats the one before, all the way to the cone. More sides is always better on that measure.

What six sides actually buys you is buildability. Six identical triangular panels, one rafter length, one bevel angle, and a base you can lay out with a compass and a straightedge. Efficiency stops improving much past a dozen sides while the carpentry gets worse with every one, and that trade, not a geometric optimum, is why gazebos come out hexagonal.

Sanity check:

• a = 0 or h = 0: V = 0. ✓
• For a = 1, h = 1: V = √3/2 ≈ 0.866. (Unit hex pyramid.)
• Doubling a quadruples V (a² scaling).