The multidimensional modal theory proposed by Faltinsen, et al. (2000) is applied to solve liquid nonlinear free sloshing in right circular cylindrical tank for the first time. After selecting the leading modes and fixing the order of magnitudes based on the Narimanov-Moiseev third order asymptotic hypothesis, the general infinite dimensional modal system is reduced to a five dimensional asymptotic modal system (the system of second order nonlinear ordinary differential equations coupling the generalized time dependent coordinates of free surface wave elevation). The numerical integrations of this modal system discover most important nonlinear phenomena, which agree well with both pervious analytic theories and experimental observations. The results indicate that the multidimensional modal method is a very good tool for solving liquid nonlinear sloshing dynamics and will be developed to investigate more complex sloshing problem in our following work.