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In this paper, we consider the design of planar arrays that optimize direction-of-arrival (DOA) estimation performance. We assume that the single-source DOA is a random variable with a known prior probability distribution, and the sensors of the array are constrained to lie in a region with an arbitrary boundary. The Crame´r-Rao Bound (CRB) and the Fisher Information Matrix (FIM) for single-source DOA constitute the basis of the optimality criteria. We relate the design criteria to a Bayesian CRB criterion and to array beamwidth; we also derive closed-form expressions for the design criteria when the DOA prior is uniform on a sector of angles. We show that optimal arrays have elements on the constraint boundary, thus providing a reduced dimension iterative solution procedure. Finally, we present example designs.