Derivative relationships between volume and surface area of compact regions in R^d

We explore the idea that the derivative of the volume, V, of a region in R^d with respect to r equals its surface area, A, where r = d V/A. We show that the families of regions for which this formula for r is valid, which we call homogeneous families, include all the families of similar regions. We determine equivalent conditions for a family to be homogeneous, provide examples of homogeneous families made up of non-similar regions, and offer a geometric interpretation of r in a few cases.