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Polyhedron with eight triangular faces

In geometry, an octahedron (: octahedra or octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is also a rectified tetrahedron, a square bipyramid in any of three orthogonal orientations, and a triangular antiprism in any of four orientations. An octahedron is the three dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3 ball in the Manhattan (ℓ) metric.

Using the standard nomenclature for Johnson solids, an octahedron would be called a square bipyramid.

The octahedron is the dual polyhedron of the cube. If an octahedron of edge length is inscribed in a cube, then the length of an edge of the cube .

The interior of the compound of two dual tetrahedra is an octahedron, and this compound, called the stella octangula, is its first and only stellation. Correspondingly, a regular octahedron is the result of cutting off from a regular tetrahedron, four regular tetrahedra of half the linear size (i. e. rectifying the tetrahedron). The vertices of the octahedron lie at the midpoints of the edges of the tetrahedron, and in this sense it relates to the tetrahedron in the same way that the cuboctahedron and icosidodecahedron relate to the other Platonic solids.

One can also divide the edges of an octahedron in the ratio of the golden mean to define the vertices of an icosahedron. This is done by first placing vectors along the octahedron's edges such that each face is bounded by a cycle, then similarly partitioning each edge into the golden mean along the direction of its vector. There are five octahedra that define any given icosahedron in this fashion, and together they define a regular compound. An icosahedron produced this way is called a snub octahedron.

Octahedra and tetrahedra can be alternated to form a vertex, edge, and face uniform tessellation of space. This and the regular tessellation of cubes are the only such uniform honeycombs in 3 dimensional space.

The octahedron is 4 connected, meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4 connected simplicial well covered polyhedra, meaning that all of the maximal independent sets of its vertices have the same size. The other three polyhedra with this property are the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron with 12 vertices and 20 triangular faces.

The regular octahedron has eleven arrangements of nets.

Especially in roleplaying games, this solid is known as a "d8", one of the more common polyhedral dice. If each edge of an octahedron is replaced by a one ohm resistor, the resistance between opposite vertices is ohm, and that between adjacent vertices ohm. Six musical notes can be arranged on the vertices of an octahedron in such a way that each edge represents a consonant dyad and each face represents a consonant triad; see hexany.

A space frame of alternating tetrahedra and half octahedra derived from the Tetrahedral octahedral honeycomb was invented by Buckminster Fuller in the 1950s. It is commonly regarded as the strongest building structure for resisting cantilever stresses.

As a trigonal antiprism, the octahedron is related to the hexagonal dihedral symmetry family.

Truncation of two opposite vertices results in a square bifrustum. The octahedron can be generated as the case of a 3D superellipsoid with all exponent values set to 1.