As an extension of the NDIF method developed by the authors, a practical analytical method for the free vibration analysis of a simply supported polygonal plate with arbitrary shape is proposed. Especially, the method is more effective for plates highly concave shapes because it employs a sub-domain method dividing the plate of interest with two sub-plates. The approximate solution of each sub-plate is assumed by linearly superposing plane waves propagated from edges of the sub-plate. Sub-system matrix equations for the two sub-plates are extracted by applying the simply supported boundary condition to the edges of each sub-plate (excepting the common interface of the two sub-plates). Finally, the sub-system matrix equations is merged into a single system matrix equation for the entire plate by considering the compatibility condition that the two sub-plates have the same displacement and slope at the common interface. The eigenvalues and mode shapes of the single plate are obtained from the determinant of a system matrix extracted from the entire system matrix equation. It is shown by several case studies that the proposed method has a good convergence characteristics and yields accurate eigenvalues and mode shapes, compared with another analytical method (NDIF method) and FEM (NASTRAN).