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13 Polyhedra

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 .

Next: 13.1 Regular Polyhedra
Up: Part II: Three-Dimensional Geometry
Previous: 12.3 Lines