A new method for calculation of the radial distribution functions (RDFs) at contact from a specified hard-sphere mixture equation of state (EOS) is proposed. The method is applied to two available accurate EOS expressions and new analytical formulas for the RDFs are obtained for binary additive hard-sphere mixtures. The results of the new formulas are compared with available computer simulation data and with those of other RDF expressions from the literature. The evidence to date suggests that the new formula is more accurate than alternative formulas currently available.