The shape of the Fermi surface of topological nodal-ring semimetals at low carrier concentrations is characterized by the ring radius b/ℏvF. This peculiar topological property may not have a clear signature in measurable physical quantities. We demonstrate an accurate and definitive method to determine the radius of topological nodal-ring semimetals. Under a magnetic field along the ring axis, the axial magneto-optical response (σzz) has a giant peak. The position of this ultrastrong response is at the frequency of exactly 2b and is independent of the strength of the magnetic field. The amplitude of the peak response is many times stronger than that of any other inter-Landau level transitions if the magnetic energy is greater than b and is similar strength if b is greater than the magnetic energy. The origin of the ultrastrong response is that the axial magnetic transition is governed by selection rules completely different to those governing σxx where the giant response is absent [R. Y. Chen et al., Phys. Rev. Lett. 115, 176404 (2015)]. The present work provides a method to accurately determine parameters of the topological properties of semimetals.