Optimum dimensions of circular fins of trapezoidal profile with variable thermal conductivity and heat transfer coefficients are obtained. Linear variation of the thermal conductivity is considered of the form k = k0(1 + εT/T0), and the heat transfer coefficient is assumed to vary according to a power law with distance from the bore, expressed as h = K[(r − r0)/(r0 − re)]m. The results for m = 0, 0.8, 2.0, and −0.4 ≤ ε ≤ 0.4, have been expressed by suitable nondimensional parameters which are presented graphically. It is shown that considering the thermal conductivity as constant, the optimum base thickness and volume of the fin are inversely proportional to the thermal conductivity of the material of the fin, while the optimum length and effectiveness are independent of the properties of the material used.