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

Derivative Relationships between Volume and Surface Area of Compact Regions in ${\bf R}^d$