An algorithm for the calculation of the volume of polyhedral ions has been worked out. It is based on the partition of polyhedra into disjoint simplexes (triangular pyramids). The algorithm allows us to perform state-of-the-art calculations on the volumes of carbon and boron-nitride fullerenes and their ions. The general dependence of the volume of fullerene ions on their charge has been determined and described by quadratic correlations. Upon ion formation, carbon fullerenes (C₆₀ and C₇₀), obeying the isolated pentagon rule, show no dependence of the volume's change on the type of fullerene. Using this fact, we have predicted the volumes of giant C₅₄₀ fullerene ions. Our approach is applicable to cage ions of another type that has been proved by analogous calculations of the volumes of Ge₉^z− ions and Fe₄S₄ clusters of the active sites of a ferredoxin protein. The present theoretical study demonstrates that general regularity of charges and volumes of nanoscale cages exists. It should be taken into account in the design of nanodevices and nanomaterials.