octaëder

An octaëder, also known in English as an octahedron, is a polyhedron with eight faces. A regular octahedron has eight equilateral triangle faces, six vertices, and twelve edges. It is one of the five Platonic solids and is the dual polyhedron of the cube, meaning its vertices correspond to the cube’s faces and its faces correspond to the cube’s vertices. The solid is also known by regional spellings such as octaëder (Dutch) or octaèdre (French).

A common construction is two square pyramids joined at their bases, yielding a symmetric form with four faces meeting at each vertex. A standard coordinate model places the vertices at (±1,0,0), (0,±1,0), and (0,0,±1). In this model the edge length is √2; scaling yields a general edge length a. Key measures for a regular octahedron with edge length a include: volume V = (1/3)√2 a^3, surface area A = 2√3 a^2, circumradius R = a/√2, and inradius r = a√6/6. The dihedral angle between adjacent faces is arccos(-1/3) ≈ 109.47°.

Symmetry is highly regular: the orientation-preserving rotational symmetry group has order 24, and the full symmetry group (including reflections) has order 48. The octahedron is closely related to crystallography and chemistry, where an octahedral coordination geometry describes certain molecular arrangements. In games, the eight-faced die used in many tabletop RPGs is usually a regular octahedron.

The name varies by language, but all refer to the same geometric object. In Dutch one writes octaëder, in French octaèdre, and in English octahedron.