  We identify a length scale that simultaneously accounts for the observed
proton-nucleus reaction cross section and diffraction peak in the proton
elastic differential cross section. This scale is the nuclear radius, $a$,
deduced from proton elastic scattering data of incident energies higher than
$\sim800$ MeV, by assuming that the target nucleus is a ``black'' sphere. The
values of $a$ are determined so as to reproduce the angle of the first
diffraction maximum in the scattering data for stable nuclei. We find that the
absorption cross section, $\pi a^2$, agrees with the empirical total reaction
cross section for C, Sn, and Pb to within error bars. This agreement persists
in the case of the interaction cross section measured for a carbon target. We
also find that $\sqrt{3/5}a$ systematically deviates from the empirically
deduced values of the root-mean-square matter radius for nuclei having mass
less than about 50, while it almost completely agrees with the deduced values
for $A\gtrsim50$. This tendency suggests a significant change of the nuclear
matter distribution from a rectangular one for $A\lesssim50$, which is
consistent with the behavior of the empirical charge distribution.
