# 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.

## 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:

$$\pm \frac{1}{6}\left|\begin{array}{cccc}\hfill {a}_{x}\hfill & \hfill {a}_{y}\hfill & \hfill {a}_{z}\hfill & \hfill 1\hfill \\ \hfill {b}_{x}\hfill & \hfill {b}_{y}\hfill & \hfill {b}_{z}\hfill & \hfill 1\hfill \\ \hfill {c}_{x}\hfill & \hfill {c}_{y}\hfill & \hfill {c}_{z}\hfill & \hfill 1\hfill \\ \hfill {d}_{x}\hfill & \hfill {d}_{y}\hfill & \hfill {d}_{z}\hfill & \hfill 1\hfill \end{array}\right|.$$ |

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 |