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For any polyhedron topologically equivalent to a sphere---in
particular, for any **convex polyhedron**---the **Euler formula**
holds:
*v*-*e*+*f*=2,
where *v* is the number of vertices, *e* is the number of edges, and
*f* is the number of faces.

Many common polyhedra are particular cases of cylinders
(Section 13.2) or cones (Section 13.3). A cylinder
with a polygonal base (directrix) is called a **prism**. A cone with
a polygonal base is called a **pyramid**. A frustum of a cone with a
polygonal base is called a **trucated pyramid**.
Formulas (13.2.1)
, (13.3.1)
and
(13.3.2)
give the volume of a general prism, pyramid,
and trucated pyramid.

A prism whose base is a parallelogram is a **parallelepiped**. The
**volume** of a parallelepiped with one vertex at the origin and
adjacent vertices at (*x*,*y*,*z*), (*x*,*y*,*z*) and
(*x*,*y*,*z*) is given by

The **rectangular parallelepiped** is a particular case: all its
faces are rectangles. If the side lengths are *a*, *b*, *
c*, the **volume** is *abc*, the **total area** is 2(*
ab*+*ac*+*bc*), and each **diagonal** has length
. When *a*=*b*=*c* we get the **
cube**: see Section 13.1.

A pyramid whose base is a triangle is a **tetrahedron**. The
**volume** of a tetrahedon with one vertex at the origin and
the other vertices at (*x*,*y*,*z*), (*x*,*y*,*z*) and
(*x*,*y*,*z*) is given by

In a tetrahedron with vertices *P*, *P*, *P*, *P*, let
be the distance (edge length) from to . Form the
determinants

Then the **volume** of the tetrahedron is , and
the radius of the **circumscribed sphere** is .

*Silvio Levy
Wed Oct 4 16:41:25 PDT 1995*

This document is excerpted from the 30th Edition of the *CRC Standard Mathematical Tables and Formulas* (CRC Press). Unauthorized duplication is forbidden.