Definition: Def. 11.26: Octahedron

An octahedron is a solid figure contained by eight equal and equilateral triangles.

Modern Formulation

An octahedron is (the only)1 regular three-dimensional polyhedron with \(8\) faces, \(12\) edges, and \(6\) vertices.

Cartesian Coordinates of the Vertices and Faces of an Octahedron

The Cartesian coordinates \((x,y,z)\) of all \(6\) vertices of an octahedron centered at the origin are given by

\[\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_{1}&1&0&0\\ v_{2}&0&1&0\\ v_{3}&0&0&1\\ v_{4}&-1&0&0\\ v_{5}&0&-1&0\\ v_{6}&0&0&-1\\ \end{array}\]

The \(8\) faces of the octahedron are equilateral triangles with the following vertices:

\[\begin{array}{lccccc} \text{Face}\\ f_{1}&v_{1}&v_{2}&v_{3}\\ f_{2}&v_{1}&v_{2}&v_{6}\\ f_{3}&v_{1}&v_{3}&v_{5}\\ f_{4}&v_{1}&v_{5}&v_{6}\\ f_{5}&v_{2}&v_{3}&v_{4}\\ f_{6}&v_{2}&v_{4}&v_{6}\\ f_{7}&v_{3}&v_{4}&v_{5}\\ f_{8}&v_{4}&v_{5}&v_{6}\\ \end{array}\]

Problems: 1
Proofs: 2 3
Propositions: 4 5
Sections: 6

Thank you to the contributors under CC BY-SA 4.0!



Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki",, 2016


  1. This will be proven in the Prop. 18 of Book 13, thus the octahedron is well-defined.