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 — 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’s roughly 370 cubic feet of conditioned air space — easy for a small portable AC or heater to manage.

Worked example — modern architectural pyramidal roof: The Louvre Pyramid (Paris) is a square pyramid, but hexagonal variants exist in some convention centers. A hexagonal roof with a = 8 m, h = 5 m: V = 0.866 × 64 × 5 ≈ 277 m³.

If insulating this with foam (R = 30 insulation): the surface area scales differently from volume, so insulation is sized by surface area, not volume.

Where hexagonal pyramids appear:

• Garden pavilions and gazebos. Six-sided ones are common in formal gardens — more interesting than square, easier to build than octagonal.
• 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

The more sides, the more volume — for fixed edge length. This makes sense: hexagonal base area is much bigger than triangular base area for the same edge.

For the same FOOTPRINT (inscribed circle), the comparison is closer — all the regular polygons approach a circle as sides increase, so their pyramids approach the same volume.

Volume vs. surface area trade-off:

Hexagonal pyramids have the BIGGEST volume for a given total surface area of any regular-base pyramid up to 6 sides — that’s why honeycomb-style structures are so efficient. Beyond 6 sides, the gains diminish; beyond 12, you’re basically dealing with a cone.

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).