The electronic properties of Na16, Cu16, and Si 20-28 clusters were calculated using density functional theory with the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof. These clusters are special, as transitions in cluster shape occur at these sizes in the Nan, Cun, and Sin cluster systems, respectively. Low-energy isomers that are comparable in stability, but possess distinctly different shapes, exist at each of these sizes, making these sets of isomers useful as probes of geometrical effects on cluster properties. Results for ionization potentials, electron affinities, and polarizabilities are shown to have a characteristic dependence on cluster shape. An analysis of the results reveals a close relationship between polarizability and cluster volume for all the isomers studied, despite the differences in cluster type and geometry. This relationship accounts for variations in polarizabilities among isomers of the same size, but different shapes, whereas previously published rules relating the polarizability to other cluster properties do not.