Abstract: The exciton states in a strained (In,Ga)As/GaAs nanocup are theoretically determined. We explore how the nanocup bottom thickness (t) affects the magnetic field dependence of the exciton energy. Strain distribution is computed by the continuum mechanical model under the approximation of isotropic elasticity. The exciton wave functions are expanded into products of the electron and hole envelope functions. For small t, the exciton ground state has zero orbital momentum and exhibits small oscillations of the second derivative when the magnetic field increases. When t approaches the value of the cup height, however, the exciton levels exhibit angular momentum transitions, whose behavior is similar to that for type-II quantum dots. Small oscillations of the oscillator strength for exciton recombination are found when the magnetic field increases. An increase in thickness of the nanocup bottom has only a small effect on those oscillations for the optically active exciton states, but the exciton ground state becomes dark when the magnetic field increases. Hence, the results of our calculations show that an increase in thickness of the nanocup bottom transforms the exciton ground energy level dependence on magnetic field from the one characteristic of type-I rings to the one characteristic of type-II dots.