The canonical inverse source problem of reconstructing an unknown source whose region of support is describable as a spheroidal (oblate or prolate) volume from knowledge of the far-field radiation pattern it generates is formulated and solved within the framework of the inhomogeneous scalar Helmholtz equation via a linear inversion framework in Hilbert spaces. Particular attention is paid to the analysis and computer illustration of flat, aperture-like sources whose support is approximated by an oblate spheroidal volume.