"Inradius" means radius of largest sphere that is tangent to all faces.
For example:
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 $A=rs$.
The 2-D analogue works because you can draw edges from the incenter to the sides of a polygon, cutting it into triangles. These triangles have a height of $r$ and their bases sum to the perimeter, so their total area is half the inradius times the perimeter.
In 3-D, you can do the same thing. By connecting the incenter to the edges of a polyhedron, you cut the polyhedron into pyramids which all have a height of $r$ and whose bases make up the surface area of the polyhedron. The volume of a pyramid is the height times one-third the area of the base, so the total volume is the inradius times one-third of the surface area. This works for any polyhedron that has an inscribed sphere touching all of its faces.
