We present calculations of structural features and magnetic field effects on the properties of an exciton localized in nanometer-size bidimensional quantum rings and disks. To analyze the effects of the magnetic field, the electron tunneling through the core and the leakage of the electron wave function toward the inner and outer barrier regions on the exciton ground state in InAs quantum rings, we consider a model of parabolic confinement potential with different barrier heights in the core and in the external surrounding environment. The exciton trial wave function is taken as a product of the unbound electron and hole wave functions in the quantum system, with an arbitrary correlation function that depends only on electron-hole separation. The uncoupled electron and hole probability densities show a high sensibility to variation of the ring and core widths. In particular, the wave functions leakage in the inner barrier region as the ring width is very small while for wide rings the effect of tunneling through this region is significant as the core radius is small. The binding energy, electron-hole separation, and the oscillator strength of the exciton are calculated as functions of the width of the ring and magnetic field strength.