Based on constructal theory, the constructal optimization of a disc on micro and nanoscales is carried out by taking minimum entransy dissipation rate as optimization objective; and the optimal construction of the disc with minimum dimensionless equivalent thermal resistance is obtained. The result shows that the optimal construction of the disc when the size effectis taken into account is obviously different from that without considering the size effect. There exists an optimal dimensionless channel length of the high conductivity material which leads to the minimum dimensionless equivalent thermal resistance. With the increase in the number of the elemental sectors, the minimum dimensionless equivalent thermal resistance decreases first and then increases, and there exists an optimal number of the elemental sectors which leads to the double minimum dimensionless equivalent thermal resistance, which is different from the performance characteristic of the disc on a conventional scale. The entransy dissipation rate of the disc, based on the minimization of entransy dissipation rate, is reduced by 7.31% as compared with that based on maximum temperature difference, that is, the average heat transfer temperature difference of the disc is reduced by 7.31%. The optimal construction on micro and nanoscales, obtained based on minimum entransy dissipation rate, can reduce the average heat transfer temperature difference of a disc, and improves its global heat transfer performance simultaneously. The work in this paper can help to further extend the application range of the entransy dissipation extremum principle.