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Octahedron Calculator

Find the volume of a regular octahedron.

How it is calculated

A regular octahedron has eight identical equilateral-triangle faces and all twelve edges of length a. The volume is V = (√2 ÷ 3) × a³ and the surface area is S = 2√3 × a². Enter the edge length a, above zero, in any length unit; the volume then comes out in that unit cubed and the surface in that unit squared. The tool labels them "units³" and "units²" since no unit is assumed. Both results are rounded to two decimals. It applies only to the regular octahedron; irregular ones have different formulas.

Worked example

Edge a = 3. Volume: a³ = 27, √2 ÷ 3 ≈ 0.4714, so V = 0.4714 × 27 ≈ 12.73. Surface: a² = 9, 2√3 ≈ 3.4641, so S = 3.4641 × 9 ≈ 31.18. The tool shows a volume of 12.73 units³ and a surface of 31.18 units². In centimetres, that is 12.73 cm³ and 31.18 cm².

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Common mistakes

  • Entering the distance between opposite corners instead of the edge length. The input is the length of one edge.
  • Mixing area and volume units. Volume is cubed (cm³), surface is squared (cm²); the labels say units³ and units².
  • Doubling the edge and expecting double the volume. Volume grows with the cube, so a doubled edge gives eight times the volume and four times the surface.
  • Using the formula for a pyramid. The octahedron is two square pyramids base to base, so it has twice the volume of one of them.
  • Applying it to a non-regular shape with unequal edges. The tool assumes all edges are equal.

About this calculator

A regular octahedron is one of the five Platonic solids, built from eight equilateral triangular faces, twelve edges and six vertices where four faces meet. Its volume follows V = (√2/3)·a³ and its surface area is 2√3·a², both derived directly from the edge length a since every face and dihedral angle is identical by symmetry; the solid can also be seen as two square pyramids joined base to base.

Crystallographers and mineralogists use octahedral geometry to describe natural crystal habits such as fluorite, spinel, magnetite and diamond, which often grow as octahedra due to their underlying cubic lattice symmetry. Chemists apply the same shape to describe octahedral molecular geometry in coordination complexes, while game designers and 3D modelers use it for eight-sided dice and low-polygon assets.

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Comments (1)

  • Editorial team

    A quick sanity check: a regular octahedron whose corners sit on the face centres of a cube fills exactly one sixth of that cube. For an edge of 3, the enclosing cube has an edge of 3 × √2, about 4.243, a volume near 76.37, and one sixth of that is 12.73, matching the tool. Do you use octahedra for dice, crystals or geometry homework?

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