So if you're in that distinct ethnic minority of being focussed on spatial geometry (so-called "solid"), not just the flat stuff, you might want to pick up a few clues for some interesting research projects, from this distillation of relatively recent results.
First, if you know what a tetrahedron is, a regular one, then consider that your "water cup" for pouring into other shapes, measuring their volume. Vis-a-vis this approach, a class of polyhedra called the Waterman Polyhedra all have whole number volumes. Use the Internet to find out more.
Second, consider the space-filling rhombic dodecahedron, the encasement for each ball in a dense-packing we call the CCP and/or FCC (other things). Given those balls are unit radius, with four of them defining our regular tetrahedron (above), this rhombic dodecahedron has a volume of six. Use your knowledge of geometry and algebra to verify this claim.
Third, consider the rhombic triacontahedron, yes a quasi- spherical shape of 30 diamond faces. Inscribed about a sphere, such that its 30 face centers kiss the sphere's surface, we define its radius as equal to that of the encased sphere's.
Verify that if this sphere has a radius of phi/sqrt(2), that this shape has a volume of 7.5, relative to our basic 'water cup' (above). This is not such an easy problem, answer tomorrow (you don't have to look).
Kirby Urner 4dsolutions.net
Note: 'Connections: The Geometric Bridge Between Art and Science' (Jay Kappraff, NJIT), is a good source of information on both phi and sqrt(2), in terms of their geometric significance and appearance in computations. phi = (1 + sqrt(5))/2 and is known as the "golden mean" or "golden proportion" (pronounced fee, fie... or some use letter the greek letter tau). phi/sqrt(2) = about 1.1441228.