Constructal theory is a means for optimization of resistances in the area-point flows. In the present study, the problem of cooling an adiabatic disc by highly conductive inserts was examined, analytically. This work was accomplished using a fixed amount of highly conductive material distributed as incomplete inserts from the heat sink at the center toward the perimeter.Using constructal theory, thermal resistances of the radial and tributary configurations were evaluated, analytically. Firstly, it was tried to derive an equation for thermal resistance of the disc for radial configuration of inserts based on the procedure implemented in constructal studies which have been done, previously. This was carried out by implementing the optimized thermal resistance of elemental sectors. Then, the computed elemental sectors were put together so that they make branching configuration of inserts in the disc while the thermal resistance of the disc by such architecture of inserts was calculated. The thermal resistance was minimized regarding to the aspect ratio to find the optimal number of inserts and the radius of the disc. From this work, It was concluded that for a specific range of R 1 /R 2 , the optimal total thermal resistance for incomplete inserts has considerably better performance in comparison with complete ones in the radial pattern. In the present study, conductive cooling of a disc was also done by means of incomplete constant and variable cross-section conducting paths embedded in radial and tributary configurations. Variational calculus was invoked to determine the optimum shape of the cross-sections of the inserts. Out of the comparison between the obtained thermal resistances of the disc with constant and variable cross-sections, it was concluded that using variable cross-sections reduces thermal resistance, but this effect differs in radial and tributary configurations, i.e., increasing the complexity of tributary patterns does not always reduce the thermal resistance more effectively in comparison with radial configurations. Moreover, the thermal conductivities were considered temperature-dependent which were varied linearly and ascending. Applying the Kirchhoff transformation, the resulting nonlinear partial equations were transformed to linear ones which would be easier to solve, analytically. Furthermore, the effect of temperature-dependency of the thermal conductivities was examined and compared with constant thermal conductivities case under different conditions where a decrease in thermal resistance was observed. Finally,a numerical solution was performed to validate the analytical results where an acceptable consistency was observed. KeyWords : Constructal, Highly conductive, Incomplete insert, Variable cross-section, Variational, Temperature-dependency, Kirchhoff transformation, Cooling, Disc.