The optimum dimensions of trapezoidal profile radiating and convective-radiating circular fins have been determined. A step-by-step derivation of the governing equations is given, which are set forth in a non-dimensional form and solved numerically. Consideration is given to both zero and non-zero environmental temperature and environmental irradiation. The analysis is based on the following simplifying assumptions: one-dimensional conduction, insulated tip, length of arc idealization ( LAI), and negligible fin-to-base radiant interaction. The influence of the thermal conductivity, emissivity and other parameters upon the optimum dimensions has been analyzed. The results are generalized by expressing the optimum bore thickness, tip radius, heat dissipation, and volume in dimensionless form. Simple correlations have been derived that can be employed to readily obtain the optimum fin dimensions for a specified heat transfer rate or fin's weight without the use of any graphs and/or interpolations. Two examples are given that serve to demonstrate the usefulness of this analysis.