We have measured for the first time the charge-changing cross sections ($sigma_{text{CC}}$) of $^{12-16}$C on a $^{12}$C target at energies below $100A$ MeV. To analyze these low-energy data, we have developed a finite-range Glauber model with a global parameter set within the optical-limit approximation which is applicable to reaction cross section ($sigma_{text{R}}$) and $sigma_{text{CC}}$ measurements at incident energies from 10$A$ to $2100A$ MeV. Adopting the proton-density distribution of $^{12}$C known from the electron-scattering data, as well as the bare total nucleon-nucleon cross sections, and the real-to-imaginary-part ratios of the forward proton-proton elastic scattering amplitude available in the literatures, we determine the energy-dependent slope parameter $beta_{rm pn}$ of the proton-neutron elastic differential cross section so as to reproduce the existing $sigma_{text{R}}$ and interaction-cross-section data for $^{12}$C+$^{12}$C over a wide range of incident energies. The Glauber model thus formulated is applied to calculate the $sigma_{text{tiny R}}$s of $^{12}$C on a $^9$Be and $^{27}$Al targets at various incident energies. Our calculations show excellent agreement with the experimental data. Applying our model to the $sigma_{text{tiny R}}$ and $sigma_{text{tiny CC}}$ for the neutron-skin $^{16}$C nucleus, we reconfirm the importance of measurements at incident energies below $100A$ MeV. The proton root-mean-square radii of $^{12-16}$C are extracted using the measured $sigma_{text{CC}}$s and the existing $sigma_{text{R}}$ data. The results for $^{12-14}$C are consistent with the values from the electron scatterings, demonstrating the feasibility, usefulness of the $sigma_{text{CC}}$ measurement and the present Glauber model.
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