  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.
