Definition: Def. 11.12: Pyramid, Tetrahedron

A pyramid is a solid figure, contained by planes, (which is) constructed from one plane to one point.

Modern Formulation

A pyramid is a three-dimensional [polyhedronsolid figure with a designated face called the base of the pyramid and one designated vertex, which is not on the base, called the top of the pyramid. All other faces of the pyramid are constructed as triangles between the sides of the base and the top. Depending on the number \(n\) of the vertices of the base, the pyramid will have \(n+1\) faces (i.e. \(n\) triangular faces and one base), \(n+1\) vertices (i.e. \(n\) vertices of the base and one top) and \(2n\) edges (i.e. \(n\) edges of the base and \(n\) edges from drawn from the vertices of the base to the top).

A tetrahedron is (the only)1 regular pyramid.

Cartesian Coordinates of the Vertices and Faces of a Tetrahedron

The Cartesian coordinates \((x,y,z)\) of all \(4\) vertices of a tetrahedron centered at the origin are given by:

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

The \(4\) faces of the tetrahedron are equilateral triangles with the following vertices:

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

Corollaries: 1 2
Problems: 3 4
Proofs: 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Propositions: 21 22 23 24 25 26 27 28 29 30
Sections: 31

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 tetrahedron is well-defined.