Materials with large birefringence (Δn) are highly needed by fiber-optic isolators, whereas crystals showing strong second-order harmonic generation (SHG) are the key component for all-solid-state laser devices. Cyanurate constructed by the planar π-conjugated ring (C3N3O3, CY) is a class of fascinating materials exhibiting not only very large Δn (larger than that of calcite) but also strong SHG (2 times that of β-BaB2O4, BBO). Here, we report five new cyanurates (I-V) and their single-crystal structures; among them, II realizes a Δn = 0.4, the maximum in the cyanurate family, and IV realizes a d33 = 6.69 pm/V, one of the highest values in the cyanurate family. We discover a dependence between Δn and the coplanarity of the CY rings that is explicitly described by a Boltzmann function, in which the coplanarity is defined by the dihedral angle (γ) between the CY plane and the principal optical axes XY plane. II realizes the maximum Δn due to its zero γ that indicates perfect coplanarity. Such a Δn-γ dependence also allows the Δn prediction of cyanurates. Furthermore, we uncover that the SHG intensity of cyanurates increases monotonically as the angle (θ) between the maximum hyperpolarizability (βmax) vector and the crystal 21 polar axis decreases. We predict the dij to extend well beyond such a value and to maximize at θ = 0°. Our results have significant implications for the future rational design and discovery of high-performance materials of π-conjugated and other related systems.