Octahedron
Overview

In geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, an octahedron (plural: octahedra) is a polyhedron
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

with eight faces. A regular octahedron is a Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...

composed of eight equilateral triangles, four of which meet at each vertex.

An octahedron is the three-dimensional case of the more general concept of a cross polytope.
If the edge length of a regular octahedron is , the radius
In classical geometry, a radius of a circle or sphere is any line segment from its center to its perimeter. By extension, the radius of a circle or sphere is the length of any such segment, which is half the diameter. If the object does not have an obvious center, the term may refer to its...

of a circumscribed sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...

(one that touches the octahedron at all vertices) is

and the radius of an inscribed sphere (tangent
Tangent
In geometry, the tangent line to a plane curve at a given point is the straight line that "just touches" the curve at that point. More precisely, a straight line is said to be a tangent of a curve at a point on the curve if the line passes through the point on the curve and has slope where f...

to each of the octahedron's faces) is

while the midradius, which touches the middle of each edge, is

An octahedron can be placed with its center at the origin and its vertices on the coordinate axes; the Cartesian coordinates of the vertices are then
( ±1, 0, 0 );
( 0, ±1, 0 );
( 0, 0, ±1 ).

The surface area A and the volume
Volume
Volume is the quantity of three-dimensional space enclosed by some closed boundary, for example, the space that a substance or shape occupies or contains....

V of a regular octahedron of edge length a are:

Thus the volume is four times that of a regular tetrahedron
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

with the same edge length, while the surface area is twice (because we have 8 vs. 4 triangles).
The interior of the compound
Polyhedral compound
A polyhedral compound is a polyhedron that is itself composed of several other polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram....

of two dual tetrahedra
Tetrahedron
In geometry, a tetrahedron is a polyhedron composed of four triangular faces, three of which meet at each vertex. A regular tetrahedron is one in which the four triangles are regular, or "equilateral", and is one of the Platonic solids...

is an octahedron, and this compound, called the stella octangula
Stella octangula
The stellated octahedron, or stella octangula, is the only stellation of the octahedron. It was named by Johannes Kepler in 1609, though it was known to earlier geometers...

, is its first and only stellation
Stellation
Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again...

.