Eight decades ago, Schottky proposed that the energy barrier at the metal-semiconductor interface, which now bears his name, should be compared with the difference of two surface quantities, the work function (WF) of the metal and the ionization potential of the semiconductor. This tradition of plotting and modeling the Schottky barrier height (SBH) against the metal WF has been followed ever since. However, success in general quantitative understanding of the SBH from physical principles has been limited, and empirical models are still relied upon. Here, we show that the stumbling block that has prevented a broadly applicable physical explanation of the SBH is the presence of surface dipole terms, inherently included in the traditional, Schottky-Mott styled analyses. By removing these surface contributions with the help of the recently developed neutral polyhedra theory, we show that the SBHs calculated for a very large number of epitaxial interfaces between metals and zinc-blende semiconductors are quantitatively explained from general chemical principles. Amazingly, SBHs calculated for 17 different semiconductors fit onto the same universal plot. Previously, SBHs needed to be grouped according to the semiconductor before analyses could be conducted separately for, and with empirical parameters specific to, each semiconductor. This work shows that the mechanism for SBH formation at metal-semiconductor interface is none other than the same chemistry responsible for charge distribution in molecules. There is no need for empirical modeling once the traditional beginning-point of SBH analysis is abandoned and the proposed new one is used.