In this paper, we prove uniqueness in determining a sound-soft ball or polyhedral scatterer
in the inverse acoustic scattering problem with a single incident point source wave in \R^N (N=2,3).
Our proofs rely on the reflection principle for the Helmholtz equation with respect to a Dirichlet hyperplane
or sphere, which is essentially a 'point-to-point' extension formula. The method has been adapted to
proving uniqueness in inverse scattering from sound-soft
cavities with interior measurement data incited by a single point source. The corresponding uniqueness
for sound-hard balls or polyhedral scatterers has also been discussed