by Kurt » Sun Apr 15, 2007 8:59 pm
I'm not going to do the problems (don't really have time right now) :(.
However, as a general note, a sphere can be thought of as a polyhedron with an infinite number of faces/vertices. Therefore, inscribed/circumscribed polyhedra that have more faces/vertices will have a closer volume (and surface area) to a sphere with the same "radius".
When inscribing, the "radius" applies to the vertices, so the one with the closer volume would be the one with more vertices - i.e., a cube. This would therefore be the larger one. When circumscribing, the "radius" applies to the middle of each face, so the one with the larger number of faces would have a closer volume to the sphere - which is the octahedron. So therefore the cube should have a larger volume again.
So does the cube have a larger volume in both cases? Tell me if there is a flaw in my logic. Applying this to the other problem, I get that a dodecahedron would have a larger volume both inscribed and circumscribed.
Last edited by
Kurt on Sun Apr 15, 2007 9:13 pm, edited 1 time in total.