In this study, firstly, the structural parameters and elastic constants of Rutil SnO2 at ambient and under pressures have been computed by Quantum Espresso (QE) software. Secondly, the elastic constants of Rutile SnO2 which are available in the literature was fitted to the polynomial. As a result of the analysis made, the pressure-dependence empirical equations of the elastic constants were obtained. The elastic constants of material were determined using these empirical relationships. Using the elastic constants obtained from both calculations, some polycrystalline quantities such as bulk module, Poisson's ratio, Young's module, shear module, anisotropy factor, sound velocity, elastic Debye temperature, and Vickers hardness were derived using Voight - Reuss - Hill approximation. The results obtained are compared with the present experimental and theoretical data.