A rational approximation of the pseudo-differential Dirichlet-to-Neumann operator for a circular cylinder is made using zero placement. For an acoustically hard circular cylinder, this zero corresponds to the dominant creeping-wave singularity in the residue series for the time-harmonic free-space surface Green’s function. Based on this operator, a wide-angle on-surface radiation condition (OSRC) is developed first for two-dimensional slender convex bodies. For the object scattering problem, this OSRC is accurate for wide-scattering angles relative to the surface normal. The notion of creeping waves is then used to develop a wide-angle, three-dimensional OSRC applicable to smooth convex bodies. Application to both hard and soft, prolate and oblate spheroids is presented, and excellent agreement is found in comparison with a more costly technique. © 2004 Acoustical Society of America.