Quote by skyT
wow. That sounds like alot of work.

Well not if you want to build a simple air capacitor for an experiment or demonstration, as opposed to practical use.
Is it possible to prove that both Laplace Equation and Gauss' Law can be used to get the same value of capacitance for a spherical capacitor?

The equation above for capacitance is independent of the geometry you chose. If you want to look into the derivation of capacitance from fundamental electromagnetic principals, perhaps the physics sub forum is a better place to ask, though I expect the wikipedia pages on capacitance already answer your questions.