  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.
