1 Definition

A tetrahedronMathworldPlanetmathPlanetmath is a polyhedron with four faces, which are triangles. A tetrahedron is called non-degenerate if the four vertices do not lie in the same plane. For the remainder of this entry, we shall assume that all tetrahedra are non-degenerate.

If all six edges of a tetrahedron are equal, it is called a regular tetrahedronMathworldPlanetmathPlanetmath. The faces of a regular tetrahedron are equilateral trianglesMathworldPlanetmath.

2 Basic properties

A tetrahedron has four vertices and six edges. These six edges can be arranged in three pairs such that the edges of a pair do not intersect. A tetrahedron is always convex.

In many ways, the geometryMathworldPlanetmath of a tetrahedron is the three-dimensional analogue of the geometry of the triangle in two dimensionsPlanetmathPlanetmathPlanetmath. In particular, the special points associated to a triangle have their three-dimensional analogues.

Just as a triangle always can be inscribedMathworldPlanetmath in a unique circle, so too a tetrahedron can be inscribed in a unique sphere. To find the centre of this sphere, we may construct the perpendicular bisectorsMathworldPlanetmath of the edges of the tetrahedron. These six planes will meet in the centre of the sphere which passes through the vertices of the tetrahedron.

The six planes which connect an edge with the midpointMathworldPlanetmathPlanetmathPlanetmath of the opposite edge (see what was said about edges coming in pairs above) meet in the barycentre (a.k.a. centroid, centre of mass, centre of gravity) of the tetrahedron.

3 Mensuration

Formulas for volumes, areas and lengths associated to a terahedron are best obtained and expressed using the method of determinantsMathworldPlanetmath. If the vertices of the tetrahedron are located at the points (ax,ay,az), (bx,by,bz), (cx,cy,cz), and (dx,dy,dz), then the volume of the tetrahedron is given by the following determinant:

Figure 1: A regular tetrahedron
Title tetrahedron
Canonical name Tetrahedron
Date of creation 2013-03-22 14:26:32
Last modified on 2013-03-22 14:26:32
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 15
Author rspuzio (6075)
Entry type Definition
Classification msc 51E99
Related topic StateOnTheTetrahedron
Related topic RegularTetrahedron3
Related topic Grafix
Related topic Triangle
Defines regular tetrahedron