tetrahedron

1 Definition

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

If all six edges of a tetrahedron are equal, it is called aregular tetrahedron. The faces of a regular tetrahedron areequilateral triangles.

2 Basic properties

A tetrahedron has four vertices and six edges. These six edges can bearranged in three pairs such that the edges of a pair do notintersect. A tetrahedron is always convex.

In many ways, the geometry of a tetrahedron is the three-dimensionalanalogue of the geometry of the triangle in two dimensions. Inparticular, the special points associated to a triangle have theirthree-dimensional analogues.

Just as a triangle always can be inscribed in a unique circle, so tooa tetrahedron can be inscribed in a unique sphere. To find the centreof this sphere, we may construct the perpendicular bisectors of theedges of the tetrahedron. These six planes will meet in the centre ofthe sphere which passes through the vertices of the tetrahedron.

The six planes which connect an edge with the midpoint of the oppositeedge (see what was said about edges coming in pairs above) meet in thebarycentre (a.k.a. centroid, centre of mass, centre of gravity) of thetetrahedron.

3 Mensuration

Formulas for volumes, areas and lengths associated to a terahedron arebest obtained and expressed using the method of determinants. If thevertices of the tetrahedron are located at the points $(a_x,a_y,a_z)$, $(b_x,b_y,b_z)$, $(c_x,c_y,c_z)$, and $(d_x,d_y,d_z)$,then the volume of the tetrahedron is given by the followingdeterminant:

Figure 1: A regular tetrahedron