One of the most important effects of complex part geometry is that the available entrance and exit angles for ultrasound are limited. We will present a study of the Inverse Born Approximation in which we have data for incident (and exit) directions confined to a conical aperture. Modeling the direct problem by the Born Approximation, we obtained analytical results for (1) a weak spherical inclusion, and (2) a penny shaped crack (modeled by an oblate spheroid). General results are: (a) the value of the characteristic function γ is constant in the interior of the flaw, but reduced in value; (b) the discontinuity at the boundary of the flaw occurs over the “lighted” portion of the flaw; (c) this discontinuity is contrasted by a region where γ is negative; and (d) new non-physical discontinuities and non-analyticities appear in the reconstructed characteristic function. These general features also appear in numerical calculations which use as input strong scattering data from a spherical void and a flat penny shaped crack in Titanium. The numerical results can be straightforwardly interpreted in terms of the analytical calculation mentioned above, indicating that they will be useful in the study of realistic flaws. We conclude by discussing the stabilization of the aperture limited inversion problem and the removal of non-physical features in the reconstruction.