We thoroughly investigate how proton-nucleus total reaction cross sections depend on the target mass number A and the proton incident energy. In doing so, we systematically analyze nuclear reaction data that are sensitive to nuclear size, namely, proton-nucleus total reaction cross sections and differential elastic cross sections, using a phenomenological black-sphere approximation of nuclei that we are developing. In this framework, the radius of the black sphere is found to be a useful length scale that simultaneously accounts for the observed proton-nucleus total reaction cross section and first diffraction peak in the proton elastic differential cross section. This framework, which is shown here to be applicable to antiprotons, is expected to be applicable to any kind of projectile that is strongly attenuated in the nucleus. On the basis of a cross-section formula constructed within this framework, we find that a less familiar A(1/6) dependence plays a crucial role in describing the energy dependence of proton-nucleus total reaction cross sections.