A solid growth method (SGM) and a dual solid method (DSM), both recently developed, are each used to predict solid shapes that provide minimal total (conduction + radiation) resistance to heat transfer in a system involving conduction in a solid whose shape is to be determined, conduction in an adjoining gas, and radiation transfer between opaque, diffuse, and gray surfaces. The performance of each method is illustrated by examining solid configurations and temperature distributions that evolve as the mass of solid is gradually increased (SGM) or reconfigured (DSM). With use of either the SGM or the DSM, the solid evolves in a manner that enhances radiation heat transfer, and it is shown that neglecting radiation in the determination of solid configurations that optimize heat transfer performance is, in general, not justified. Despite the formalism of the DSM, which is based on topological optimization, the thermal performance of the DSM only marginally surpasses that of the SGM in terms of calculated total thermal resistance values, and only for cases involving a high solid thermal conductivity. For low solid thermal conductivity cases, the SGM outperforms the DSM with the difference in performance attributed to the inability of DSM to capture the fine solid structure of the SGM predictions.