Octahedron Volume Calculator (Regular)

A regular octahedron has eight congruent equilateral triangle faces, six vertices and twelve edges. By face count it is the third of the five Platonic solids, after the tetrahedron’s 4 and the cube’s 6. Picture it as two square pyramids glued base to base.

V = (√2 / 3) × s³ ≈ 0.4714 × s³

Where s is the edge length (the same for all twelve edges).

Worked example: a d8 for tabletop gaming A standard 16 mm d8 has s = 16 mm. V = 0.4714 × 4,096 ≈ 1,931 mm³ ≈ 1.93 cm³.

At plastic density 1.2 g/cm³ that is 2.3 g per die. Exactly 4 times the volume of a d4 cut to the same edge, which is a neat coincidence: the octahedron has twice the faces but four times the content, because it is also a much rounder solid.

Where octahedra appear in real measurements:

• d8 dice (8-sided gaming dice). Standard tabletop RPG dice.
• Fluorite crystals. Naturally form perfect octahedra in mineral specimens. One of the cleanest examples of regular polyhedra in nature.
• Diamond crystals. Often form octahedral habit (though they can also be cubic or dodecahedral).
• Spinel and magnetite crystals. Common octahedral mineral specimens.
• Coordination chemistry. An octahedral complex puts six ligands at the six vertices around a central metal atom, which is the commonest arrangement in transition-metal chemistry.
• Crystallography teaching models. Plastic or wooden octahedra for chemistry and geology classes.

The two-pyramid interpretation:

A regular octahedron is exactly two identical square pyramids meeting at a shared square base. It is a useful mental model:

• Each pyramid has a square base of side s.
• Each pyramid has a slant height of √(s² − (s/2)²) = √(3s²/4) = s√3/2 from base edge midpoint to apex.
• Each pyramid has a perpendicular height of s × √(2)/2 from apex to the center plane.
• Each pyramid has volume (1/3) × s² × (s√2/2) = s³√2 / 6.
• Two pyramids: 2 × s³√2/6 = s³√2 / 3. ✓

Useful octahedron measurements (all derived from s):

Octahedron vs. cube, duals of each other

The cube and the octahedron are duals. Connect the centres of a cube’s six faces and you get an octahedron; connect the centres of an octahedron’s eight faces and you get a cube.

For dual polyhedra, the number of vertices of one equals the number of faces of the other:

• Cube: 8 vertices, 6 faces.
• Octahedron: 6 vertices, 8 faces.

This dual relationship comes up in crystallography (cubic and octahedral crystals are related by their symmetry), graph theory, and architecture.

Comparing volumes (for the same edge length):

• Cube: V = s³
• Octahedron: V ≈ 0.471 × s³
• Tetrahedron: V ≈ 0.118 × s³

For a fixed edge the octahedron holds 47% of the cube and four times the tetrahedron. Two different things are going on there, and they are easy to conflate. The cube wins on raw size because six squares of side s simply bound more space than eight triangles of side s. But the octahedron is the rounder solid of the two, scoring 0.846 on sphericity against the cube’s 0.806 and the tetrahedron’s 0.671. Bigger and rounder are not the same measurement, and the cube is the standing proof of it.

Sanity check:

• s = 0: V = 0. ✓
• s = 1: V = √2/3 ≈ 0.471. ✓