Definition: Def. 11.25: Cube

(Definition 25 from Book 11 of Euclid's “Elements”)

A cube is a solid figure contained by six equal squares.

Modern Formulation

A cube is (the only)1 regular prism (hexahedron) .

Cartesian Coordinates of the Vertices and Faces of a Cube

The Cartesian coordinates \((x,y,z)\) of all \(8\) vertices of a cube centered at the origin are given by:

\[\begin{array}{lrrr} \text{Vertex}&x&y&z\\ v_{1}&-1&-1&1\\ v_{2}&-1&-1&-1\\ v_{3}&1&-1&-1\\ v_{4}&1&-1&1\\ v_{5}&1&1&1\\ v_{6}&1&1&-1\\ v_{7}&-1&1&-1\\ v_{8}&-1&1&1\\ \end{array}\]

The \(6\) faces of the octahedron are equilateral triangles with the following vertices:

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

Corollaries: 1
Definitions: 2
Proofs: 3 4 5 6
Propositions: 7 8 9 10
Sections: 11

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 cube is well-defined.