In this paper a new Kirchhoff plate model is developed for the static analysis of isotropic micro-plates with arbitrary shape based on a modified couple stress theory containing only one material length scale parameter which can capture the size effect. The proposed model is capable of handling plates with complex geometries and boundary conditions. From a detailed variational procedure the governing equilibrium equation of the micro-plate and the most general boundary conditions are derived, in terms of the deflection, using the principle of minimum potential energy. The resulting boundary value problem is of the fourth order (instead of existing gradient theories which is of the sixth order) and it is solved using the Method of Fundamental Solutions (MFS) which is a boundary-type meshless method. Several plates of various shapes, aspect and Poissonβs ratios are analyzed to illustrate the applicability of the developed micro-plate model and to reveal the differences between the current model and the classical plate model. Moreover, useful conclusions are drawn from the micron-scale response of this new Kirchhoff plate model.