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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 .
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.