Octahedron Volume Calculator (Regular)
Compute regular octahedron volume from edge length.
For d8 dice, mineral specimens, and double-pyramid crystal forms in geology and crystallography.
A regular octahedron is the second-simplest Platonic solid. Eight congruent equilateral triangle faces, six vertices, twelve edges. Visualize 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 — d8 die 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: 2.3 g per die. Roughly 4× the volume of the d4 with the same edge length.
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.
- Pyramid-shape filing cabinets and architectural display pedestals.
- Crystallography teaching models. Plastic or wooden octahedra for chemistry and geology classes.
The two-pyramid interpretation:
A regular octahedron is exactly two square pyramids meeting at a square base — both pyramids identical. This 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):
| Quantity | Formula | Value for s = 1 |
|---|---|---|
| Edge length | s | 1 |
| Face area (equilateral triangle) | (√3 / 4) × s² | 0.433 |
| Total surface area | 2√3 × s² | 3.464 |
| Volume | s³ × √2 / 3 | 0.471 |
| Vertex-to-vertex (across) | s × √2 | 1.414 |
| Inradius (insphere) | s × √6 / 6 | 0.408 |
| Circumradius (circumsphere) | s × √2 / 2 | 0.707 |
Octahedron vs. cube — duals of each other:
The cube and octahedron are “duals” — if you connect the centers of each face of a cube, you get an octahedron. Conversely, connecting the centers of an octahedron’s faces gives 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
Octahedron holds 47% of the cube’s volume — significantly more than the tetrahedron does. This is because the octahedron is more “ball-like” — closer to a sphere in proportions.
Sanity check:
- s = 0: V = 0. ✓
- s = 1: V = √2/3 ≈ 0.471. ✓
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