  We analyze total reaction cross sections, $\sigma_R$, for exploring their
sensitivity to the neutron-skin thickness of nuclei. We cover 91 nuclei of O,
Ne, Mg, Si, S, Ca, and Ni isotopes. The cross sections are calculated in the
Glauber theory using the density distributions obtained with the
Skyrme-Hartree-Fock method in 3-dimensional coordinate space. Defining a
reaction radius, $a_R=\sqrt{\sigma_R/\pi}$, to characterize the nuclear size
and target (proton or $^{12}$C) dependence, we find an empirical formula for
expressing $a_R$ with the point matter radius and the skin thickness, and
assess two practical ways of determining the skin thickness from proton-nucleus
$\sigma_R$ values measured at different energies or from $\sigma_R$ values
measured for different targets.
