This paper shows that nuclear radii and neutron skins do directly reflect the saturation density of asymmetric nuclear matter. The proton distributions in a nucleus have been found to be remarkably independent of the equation of state (EOS) of the asymmetric matter. It is the neutron distributions that are dependent on the EOS. Macroscopic model calculations have been performed over the entire range of the nuclear chart based on two popular phenomenological, but distinctively different, EOS: the Sill parameter set for the non-relativistic Skyrme Hartree-Fock theory and the TM1 parameter set in the relativistic mean field theory. The saturation density for a small proton fraction remains almost the same as the normal nuclear matter density for Sill EOS, but it becomes significantly small for the TM1 EOS. The key EOS parameters used to describe the saturation density are the density derivative of the symmetry energy and the incompressibility of symmetric nuclear matter, while the saturation energy is written using the symmetry energy alone as a good approximation. We conclude that a systematic experimental study of heavy unstable nuclei would enable us to determine the EOS of asymmetric nuclear matter at about the normal nuclear matter density with a fixed proton fraction down to about 0.3. O 1998 Elsevier Science B.V.