# tetrahedron

## 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.

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:

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