The cascade algorithm that is used for extended surface analysis depends on a new parameterization called the thermal transmission matrix to represent a single fin. This thermal transmission matrix, which is intended to replace the more familiar fin efficiency as a design and analysis parameterization, is a linear transformation that maps conditions of heat flow and temperature at the fin tip to heat flow and temperature conditions at the fin base. The cascade algorithm was derived by resorting to an analogy between a fin and the electrical transmission line. The cascade algorithm permits a fin to be subdivided into many subfins each having a thermal transmission matrix and then the individual transmission matrices for each of the subfins can be used, via a simple matrix product to form an overall equivalent thermal transmission matrix for the entire fin. This thesis develops a thermal transmission matrix for the radiating rectangular, trapezoidal, and triangular fins both for the free space and non-free space environments. Test cases have been run and their solutions exactly match those contained in the literature. The thesis concludes with optimization studies for Thermal Transmission Matrix, Cascade Algorithm, Extended Surface, Cooling Fin, Radiative Fin, Longitudinal Fin, Rectangular Fin, Trapezoidal Fin, Triangular Fin, Optimum.