We discuss the low energy electronic states in hexagonal rings. These states correspond to the transverse modes in core-shell nanowires built of III-V semiconductors which typically have a hexagonal cross section. In the case of symmetric structures the 12 lowest states (including the spin) are localized in the corners, while the next following 12 states are localized mostly on the sides. Depending on the material parameters, in particular the effective mass, the ring diameter and width, the corner and side states may be separated by a considerable energy gap, ranging from few to tens of meV. In a realistic fabrication process geometric asymmetries are unavoidable, and therefore the particles are not symmetrically distributed between all corner and side areas. Possibly, even small deformations may shift the localization of the ground state to one of the sides. The transverse states or the transitions between them may be important in transport or optical experiments. Still, up to date, there are only very few experimental investigations of the localization-dependent properties of core-shell nanowires.