This paper establishes the performance characteristics of convective–radiative longitudinal fins of rectangular, trapezoidal and concave parabolic profiles with simultaneous variation of thermal conductivity, heat transfer coefficient and surface emissivity with temperature. The convection and radiation sink temperatures were assumed to be non-zero. The calculations are carried out using the differential transformation method (DTM). The accuracy of the DTM is confirmed by comparing its predictions with the results from an analytical solution and a well-tested numerical procedure. A new idea of volume adjusted fin heat transfer rate, fin effectiveness, and fin efficiency is introduced to compare the performances of trapezoidal and concave parabolic fins with the rectangular fin. Results presented illustrate the effects of thermal conductivity parameter, emissivity parameter, convection–conduction parameter, radiation–conduction parameter, and dimensionless convection and radiation sink temperatures on the performance of fins. ► The paper investigates the performance characteristics of convective–radiative longitudinal fins with different profiles. ► The calculations are carried out using the differential transformation method which is a well-tested analytical technique. ► The comparison is based on a new idea of volume adjusted fin heat transfer rate, efficiency, and effectiveness. ► Volume adjusted results show that the concave parabolic fin provides the highest fin efficiency and fin effectiveness. ► There is an inverse relationship between the taper ratio of the fin and its local temperature.