Hexagonal Prism Volume Calculator

A hexagonal prism has two parallel regular hexagon ends and six rectangular side faces.

V = (3√3 / 2) × a² × L ≈ 2.598 × a² × L

Where a is the edge length of the hexagon (all six sides equal) and L is the prism length. The leading constant comes from the regular hexagon area:

A_hexagon = (3√3 / 2) × a² ≈ 2.598 × a²

A regular hexagon equals six equilateral triangles meeting at the centre, and that is where the formula comes from.

Worked example: a standard wooden pencil A #2 Ticonderoga pencil has a hexagonal cross-section with edge a ≈ 3.7 mm (flat-to-flat width about 6.4 mm), L = 175 mm. A_hexagon = 2.598 × 13.69 ≈ 35.57 mm². V = 35.57 × 175 ≈ 6,224 mm³ ≈ 6.2 cm³ of wood per pencil.

At cedar wood density 0.5 g/cm³: ~3.1 g of wood. The graphite core is a small cylinder.

Worked example: a honeybee comb cell A worker bee comb cell is hexagonal, edge ≈ 2.7 mm, depth ≈ 12 mm. A_hexagon = 2.598 × 7.29 ≈ 18.94 mm². V = 18.94 × 12 ≈ 227 mm³ ≈ 0.227 cm³ per cell.

A typical honey cell holds 0.2-0.3 g of honey when full. A standard 8-frame deep box holds ~50,000 cells = ~12 kg of honey at full capacity.

Where hexagonal prisms show up everywhere:

• Pencils. Industry-standard hexagonal cross-section, and it stops the pencil rolling off a sloped desk. It does not save wood: a hexagon 6.4 mm across the flats holds 35.5 mm² of cross-section against 32.2 mm² for a round pencil of the same width, so the hex actually uses about 10% more. The real saving is upstream, in the slats. Hex blanks cut from a flat cedar slat leave almost no gap between neighbours, where round blanks leave a curved sliver of waste along every seam.
• Hex nuts and bolts. Every standard fastener, because a wrench needs flat faces to grip.
• Honeycomb cells. Bees build hexagonal cells because they enclose the most volume for the least wax, which is the optimal solution for tiling a plane.
• Hexagonal screw heads. Allen keys (hex wrenches) grip the interior of a hex socket.
• Pencil cases and storage tubes. Sometimes hex-shaped to nest pencils efficiently.
• Basalt columns in nature. Cooling lava can crack into hexagonal columns, the Giant’s Causeway in Northern Ireland being the famous example.

Why hexagons are nature’s favorite:

The hexagon is the regular polygon with the most sides that still tiles a plane without gaps. (Triangles, squares, and hexagons are the only three regular polygons that tile.) Of these three, the hexagon has the lowest perimeter-to-area ratio, so bees use the least wax per cubic millimetre of stored honey.

Engineers exploit this in honeycomb sandwich panels: aluminum or aramid sheets with a hexagonal-cell core have extreme stiffness-to-weight ratios. Used in aircraft floors, racing bicycle frames, satellite panels.

Comparing to a square prism of the same edge:

Hexagonal prism: V = 2.598 × a² × L. Square prism (same edge a, length L): V = a² × L.

A hex prism holds 2.6 times as much as a square prism of the same edge length, which is 1.6 times more, not 2.6 times more. The two phrasings get mixed up constantly and the difference is a whole prism. What you give up is packing: hexagons tile beautifully against each other but waste the corners of any rectangular crate.

Sanity check:

• L = 0: V = 0. ✓
• a = 0: V = 0. ✓
• For a = 1, L = 1: V ≈ 2.598. (Unit hex prism.)