Simultaneous effects of quantum well thickness and radius on exciton energy and density exciton states on a semiconductor nanoribbon with a finite and rectangular confining potential are studied theoretically. We show that in the structural adiabatic limit, when the width of the pattern of the particles pathways within the ribbon is much smaller than its radius, the wave equations for exciton are separable and their solutions can be found numerically. We present results of calculation of the density exciton states as functions of the wavelength and energy for different values of ribbon radius and quantum well thickness, with potential profiles rectangular and parabolic.