Volume of an Octahedron

What is a regular octahedron?

A regular octahedron is one of the five Platonic solids. It consists of 8 equilateral triangular faces, 6 vertices, and 12 equal edges. It can be visualized as two regular square pyramids joined at their common square base.

Octahedron volume formula

The volume of a regular octahedron is calculated by the formula:

$$V=\frac{\sqrt{2}}{3}\cdot a^3$$

The height of the octahedron (distance between opposite vertices):

$$H=a\sqrt{2}$$

where a is the edge length of the octahedron.

Applications of octahedron volume calculations

• Crystallography and mineralogy. The octahedral shape of natural crystals is found in diamonds, fluorite, magnetite, and alum crystals. Calculating the volume of a crystal is important for determining mineral density and analyzing crystal lattice parameters.
• Chemistry and molecular geometry. The sulfur hexafluoride (SF₆) molecule has a regular octahedral shape — a central sulfur atom surrounded by six fluorine atoms. Calculating the volume of molecular models helps analyze steric effects (spatial hindrance) and the volume of molecular cavities.
• Manufacturing and jewelry design. Octahedral parts (e.g., jewelry or decorative elements) require precise volume calculation to determine the mass of the finished product and the consumption of metal (gold, silver) or plastic during casting.
• Tabletop games and design. The eight-sided die (d8) used in role-playing games (like Dungeons & Dragons) is a regular octahedron. Calculating the volume helps manufacturers estimate plastic or resin consumption per production batch.
• Architecture and geodesic structures. Octahedral space frames are used in domes and roofing systems due to their high load-bearing capacity with minimal weight. Volume calculations of steel or concrete elements are needed to estimate the total weight of the structure.
• 3D printing and materials science. When printing octahedral models, housings, or prototype parts, knowing the volume allows slicing software to accurately estimate the amount of liquid resin or filament needed for printing and calculate production costs.

How to use the calculator

1. Enter the edge length of the octahedron a.
2. Select the required decimal precision.
3. Click "Calculate" — the volume and height will appear instantly.

FAQ

Can I use this formula for an irregular octahedron?
No. The formula V = (√2/3)·a³ applies only to a regular octahedron where all 12 edges are equal. For an irregular octahedron, you need to know the coordinates of all six vertices and apply a more complex volume formula based on tetrahedral decomposition or determinants.

How are the volume and surface area of a regular octahedron related?
Both depend only on the edge a: surface area SA = 2√3·a² grows proportional to a², while volume grows proportional to a³. Doubling the edge quadruples the area and multiplies the volume by 8.

What is the ratio of octahedron volume to a cube with the same edge?
V_octahedron / V_cube = (√2/3)·a³ / a³ ≈ 0.4714. The octahedron's volume is approximately 47% of the cube's volume with the same edge length.