A spherical acoustic wave is scattered by a bounded obstacle. A generalization of the ‘optical theorem’ (which relates the scattering cross‐section to the far‐field pattern in the forward direction for an incident plane wave) is proved. For a spherical scatterer, low‐frequency results are obtained by approximating the known exact solution (separation of variables). In particular, a closed‐form approximation for the scattered wavefield at the source of the incident spherical wave is obtained. This leads to the explicit solution of some simple near‐field inverse problems, where both the source and coincident receiver are located at several points in the vicinity of a small sphere.