What are the volumes of the regular polyhedra?
geometry/platonic.volume New
New entry, edited from material collected for the archive and not yet in a released version.
Problem
What are the volumes of the regular polyhedra? [ Editor: problem statement reconstructed from the reply; the original question, "volume of an n-gon", asked for the volume of a regular n-hedron and is not in the collected material. ]
Show solution
Here's a partial answer to your query as to the volume of a regular n-hedron. The most important thing to note is that there are only five of these beasts: tetrahedron, cube, octahedron, dodecahedron, icosahedron. This can be proved using the fact that the Euler characteristic of any polyhedron is 2. For details, see Courant and Robbins' book, "What is Mathematics." The first three are easy. The tetrahedron is just the base area times the altitude/3. The area of an equilateral triangle of side x is (x^2)3^0.5/4. The altitude is easily found by the Pythagorean theorem. The cube is obvious. The octahedron can be divided into two square pyramids. the volume of each is again base area times altitude/3, both of which are easily calculated. I've never done a dodecahedron or icosahedron, but I do remember seeing the dodecahedron formula posted in the last few weeks either in this group or sci.math. -- L. Paul Gilden <lgilden@nova.umuc.edu> [ Editor: the dodecahedron and icosahedron are not worked out in the collected postings. The "Euler characteristic of any polyhedron is 2" holds for convex polyhedra (more generally, those topologically equivalent to a sphere). ]