In this work, the ground state properties of two dimensional Fermi gas system interacting in a potential consisting of harmonic and Gaussian terms are investigated in the frame of Thomas-Fermi approximation. The depth and the curvature of the potential are changed by varying confinement parameters and the influence of the constraining conditions on the system properties like the density profile, the kinetic and the potential energy of the fermionic system is analyzed comprehensively. The deviations of the results due to the Gaussian potential are also determined by comparing the results with those obtained for pure harmonic potential. Calculations are also performed analytically for non-interacting case for comparative purposes. The results show that the confinement parameters play crucial role on the ground state properties of confined system.