tetrahedron

1 Definition

A tetrahedron 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 tetrahedron. The faces of a regular tetrahedron are equilateral triangles.

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

Just as a triangle always can be inscribed 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 bisectors 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 midpoint 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 determinants. If the vertices 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 following determinant:

$\pm\frac{1}{6}\left|\begin{matrix}a_{x}&a_{y}&a_{z}&1\\ b_{x}&b_{y}&b_{z}&1\\ c_{x}&c_{y}&c_{z}&1\\ d_{x}&d_{y}&d_{z}&1\\ \end{matrix}\right|.$

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