How can you use the measures of the interior angles of regular polygons to show that a platonic solid cannot be made from regular polygons that have more than five sides?

Answer 1

A platonic solid is a regular tetrahedron where all faces, sides and angles are congruent.
To create a solid, there must be at least three faces that meet at a vertex, which limits the interior angle of each face (polygon) to be less than 120 degrees. (At 120 degrees, the solid is a plane that contains no volume). The regular polygon with the most number of sides that satisfies this criterion is the pentagon, with an interior angle of 108 degrees (<120).

For more information and interesting reading, see:
http://en.wikipedia.org/wiki/Platonic_solid