Considering a temperature dependent heat transfer coefficient, the optimum dimensions of longitudinal rectangular fins and cylindrical pin fins have been investigated analytically. Employing the parameters of fin thickness or fin diameter and tip temperatures, the optimum dimensions, heat dissipation, and efficiency of fins are obtained by Lagrange’s multiplier method. It is shown that the fin efficiency of an optimum cylindrical pin fin is higher than that of an optimum rectangular fin. To facilitate the thermal design of the heat transfer components, very simple mathematical expressions for the optimum dimensions of fins are presented in this study.