What is an octahedron?

Worked example. An octahedron with 4 cm edges has a surface area of 2√3 × 4² = 32√3 ≈ 55.43 cm² and a volume of (√2 ÷ 3) × 4³ ≈ 30.17 cm³.

Why the volume works out that way. The octahedron is two square pyramids, so MathWorld notes its volume “is two times the volume of a square-base pyramid.” Each pyramid has a square base of a² and a height of a ÷ √2.

A number you have seen before. The angle between two faces is arccos(−1/3) ≈ 109.47°, the same angle as the bonds in a methane molecule on the tetrahedron page. Both are exactly arccos(−1/3).

Make one

The net of a octahedron

An octahedron has exactly 11 different nets, the same number as the cube, its dual. Press New net above to fold some of them.

You always cut 5 of the 12 edges. Seven edges stay as folds and 5 get cut: one less than the 6 vertices.

To build it: press Print this net, cut along the solid lines, fold along the dashed lines, put glue on the grey tabs and close it up one face at a time. Albrecht Dürer printed some of the first nets like this in his 1525 book on measuring with compass and ruler. All five Platonic solids, with every net, are in Platonic Solids 3D.

Name and history

Air, Theaetetus and the cube inside out

The name is Greek: okta, eight, and hedra, a seat or face. English has used it since the 1560s.

Plato made it air, the element between fire’s tetrahedron and water’s icosahedron. The Timaeus gives “to air that which is intermediate.”

Theaetetus first studied it. MacTutor: “Theaetetus was the first to study the octahedron and the icosahedron.” An old note quoted by Thomas Heath gives the cube, pyramid and dodecahedron to the Pythagoreans and these two to Theaetetus.

It is the cube’s dual. Put a dot in the middle of each of a cube’s 6 faces and join them: you get an octahedron. Do the same to an octahedron’s 8 faces and you get a cube. The numbers swap too: 6 faces and 8 corners against 8 faces and 6 corners.

Out in the world

Where you see a octahedron

Diamonds. The Handbook of Mineralogy describes diamond crystals as “Most commonly octahedral, crystals to 10 cm or more.” A rough diamond often looks like two pyramids base to base.

Magnetite and fluorite. Magnetite, the magnetic iron ore, is “Typically octahedral.” Fluorite grows as cubes and octahedra and splits cleanly along octahedral planes. Even salt, usually cubic, sometimes grows as octahedra.

Dice. A fair die needs identical faces, so eight-sided dice are regular octahedra.

Sacred geometry. The flat outline of an octahedron can be traced inside Metatron’s Cube, and an octahedron sits at the heart of the merkaba, where two tetrahedra overlap.

The octahedron is one of exactly five Platonic solids, and Euclid proved around 300 BCE that there can never be a sixth.