"Inradius" means radius of largest sphere that is tangent to all faces.
Cube - Surface area $= 6a^2$, Inradius $= a/2$, Volume $= a^3$.
Sphere - Surface area $= 4\pi r^2$, Inradius $= r$, Volume $= 4/3 \pi r^3$.
I know this won't work for all solids, but for most (including regular pyramids) it works.
Is there a reason why this formula works? In particular where did the 1/3 come from?
Note: I am familiar with the 2-D analogue $A=rs$.