The Neumann Equation of State (EQS) allows obtaining the value of the surface free energy of a solid ${\gamma}_{SV}$ from the contact angle $({\theta})$ of a probe liquid with known surface tension ${\gamma}_{LV}$. The value of ${\gamma}_{SV}$ is obtained by numerical methods solving the corresponding EQS. In this work, we analyzed the discrepancies between the values of ${\gamma}_{SV}$ obtained using the three versions of the EQS reported in the literature. The condition number of the different EQS was used to analyze their sensitivity to the uncertainty in the ${\theta}$ values. Polynomials fit to one of these versions of EQS are proposed to obtain values of ${\gamma}_{SV}$ directly from contact angles $({\gamma}_{SV} ({\theta}))$ of particular probe liquids. Finally, a general adjusted polynomial is presented to obtain the values of ${\gamma}_{SV}$ not restricted to a particular probe liquid $({\gamma}_{SV}({\theta},{\gamma}_{LV}))$. Results showed that the three versions of EQS present non-negligible discrepancies, especially at high values of ${\theta}$. The sensitivity of the EQS to the uncertainty in the values of ${\theta}$ is very similar in the three versions and depends on the probe liquid used (greater sensitivity at higher ${\gamma}_{LV})$ and on the value of ${\gamma}_{SV}$ of the solid (greater sensitivity at lower ${\gamma}_{SV})$. The discrepancy of the values obtained by numerical resolution of both the fifth-order fit polynomials and the general fit polynomial was low, no larger than ${\pm}0.40\,mJ/m^{2}$. The polynomials obtained allow the analysis and propagation of the uncertainty of the input variables in the determination of ${\gamma}_{SV}$ in a simple and fast way.