The stellated octahedron is the only stellation of the octahedron. It is also called the stella octangula (Latin for "eight-pointed star"), a name given to it by Johannes Kepler in 1609, though it was known to earlier geometers. It was depicted in Pacioli's 1509 De Divina Proportione.

It is the simplest of the five regular polyhedral compounds, and the only regular polyhedral compound composed of only two polyhedra.

It can be seen as a 3D extension of the hexagram: the hexagram is a two-dimensional shape formed from two overlapping equilateral triangles, centrally symmetric to each other, and in the same way, the stellated octahedron can be formed from two centrally symmetric overlapping tetrahedra. This can be generalized to any desired number of higher dimensions; the four-dimensional equivalent construction is the compound of two 5-cells.

The stellated octahedron is constructed by a stellation of the regular octahedron. In other words, it extends to form equilateral triangles on each regular octahedron's faces. It is an example of a non-convex deltahedron. Magnus Wenninger's Polyhedron Models denote this model as nineteenth W₁₉.

The stellated octahedron is a faceting of the cube, meaning removing part of the polygonal faces without creating new vertices of a cube. It has the same three-dimensional point group symmetry as the cube, an octahedral symmetry.

The stellated octahedron is also a regular polyhedron compound, when constructed as the union of two regular tetrahedra. Hence, the stellated octahedron is also called "compound of two tetrahedra". The two tetrahedra share a common intersphere in the centre, making the compound self-dual. There exist compositions of all symmetries of tetrahedra reflected about the cube's center, so the stellated octahedron may also have pyritohedral symmetry.

The stellated octahedron can be obtained as an augmentation of the regular octahedron, by adding tetrahedral pyramids on each face. This results in its volume being the sum of eight tetrahedra's and one regular octahedron's volume, ${\textstyle {\frac {3}{2}}}$ times the side length. However, this construction is topologically similar as the Catalan solid of a triakis octahedron with much shorter pyramids, known as the Kleetope of an octahedron.

It can be seen as a {4/2} antiprism; with {4/2} being a tetragram, a compound of two dual digons, and the tetrahedron seen as a digonal antiprism, this can be seen as a compound of two digonal antiprisms.