2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/131355We 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.15 pagesMetric GeometryFunctional Analysis51M25, 52A38 (Primary) 26A24, 52B60 (Secondary)text