The volume of a cube based on the volume of cylinder?? Are you sure it wasn't the other way round?
Anyway, as Hottenanny pointed out you can just calculate these kind of volume using integrals or (essentially the same) http://fr.wikipedia.org/wiki/Principe_de_Cavalieri" [Broken] (sorry, only available in French and German)
Yes, a volume of "1" is defined as the volume of a "unit" cube- that is, the volume of a cube of length "1" on each side.
The fact that the volume of an n by n cube, for n an integer, is n3 does require a proof and the fact that the volume of an x by x cube, with x any real number, is x3 requires a significantly harder proof.
Once we are "given" the area of a circle, the proof that the volume of a right circular cylinder is [itex]\pi r^2 h[/itex] is fairly simple.
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