A coupled-mode method for computing the wavefield scattered by three-dimensional, axisymmetric objects is presented. Only the compressional properties of the object and the medium are modeled. The method provides a unified approach to solving three-dimensional scattering problems for axisymmetric objects in free-space, in a waveguide, or partially or fully buried in a basement. In the cases of an infinite space, the computational domain is truncated and an artificial attenuation profile added near the boundaries to attenuate artificial reflections. Galerkin's method is used to construct the mode set in this case. Numerical examples of scattering from cylindrical disks surrounded by a homogeneous space and from disks partially buried in an underlying basement are presented. A benchmark case of scattering from an infinitely long cylinder, for which analytic solutions can be constructed, is also discussed.