The size effect obtained when studying the effective properties of plates is investigated by producing a third-order correlation approximation and Hashin-Shtrikman bounds for the effective in-plane conductivity of heterogeneous plates. The boundary condition of the plates is taken into account by obtaining the exact Green operator for the boundary problem. Results are obtained for a two-dimensional (2D) random distribution of disks and a 3D distribution of spheres. All results recover those obtained for an infinite medium when the heterogeneity size becomes small compared to plate thickness. They show that the size effect is more significant in the case of a 2D distribution of heterogeneities than for a 3D distribution.