An extension of the discrete Fourier transform (DFT)-based forward-backward algorithm is developed using the virtual-element approach to provide a fast and accurate analysis of electromagnetic radiation/scattering from electrically large, planar, periodic, finite (phased) arrays with arbitrary boundaries. Both the computational complexity and storage requirements of this approach are O(Ntot) (Ntot is the total number of unknowns). The numerical results for both printed and freestanding dipole arrays with circular and/or elliptical boundaries are presented to validate the efficiency and accuracy of this approach.