We analyze the effect of the geometrical shape and size of the cross section on the spin-polarized transport and the giant magnetoresistance (GMR) by a finite element method, and evaluate the stability and the physical properties of nano-scale spin valves. We calculate the transmission coefficients in the ballistic regime by using a transfer-matrix method, and evaluate the GMR of the current perpendicular to the plane (CPP) by using a circuit theory. The conduction-band structure is simplified to the potential step, which is determined by combining the interfacial parameters calculated by first-principles with the free electron model. The geometrical shapes of the cross section are line and square. As a result, the cross sectional shape has a significant effect on the spin-polarized transport and the GMR. The square-shaped cross section has an advantage of the large GMR, which is contrary to the line-shaped cross section. These phenomena result from the difference of the cut-off energies with the transverse modes and, consequently, the different spin-down transmission coefficients.