This paper involves the use of the Galerkin finite element method to determine the temperature distribution in a rectangular cooling fin. The governing equation is a one-dimensional second order differential equation. The result shows that the temperature at the tip of the rectangular cooling fin which was 100 0 C and begins to drop as it proceeds to the other end of the rectangular cooling fin which is 61.5518 0 C at 0.1m. The result obtained from the finite element solutions when compared with the analytical solution, shows that the accuracy was very high with the highest percentage error of 0.000432875. It can be stated that the finite element solution is an accurate method for determining the temperature distribution in a rectangular cooling fin.