A geometrical theory of diffraction (GTD) analysis of the principal plane far-field radiation patterns of a hyperboloidal sub-reflector with a conical flange attachment (HWF) fed by a primary feed located at its focus is presented. While using the uniform geometrical theory of diffraction (UGTD) for evaluating the nonaxial fields, the method of equivalent currents is used in the axial region. In this paper, both the diffraction by the wedge formed between the hyperboloid and the conical flange and the diffraction by the edge of the flange are considered. While considering the diffraction by the edge due to the diffracted ray from the wedge in the H-plane, the slope diffraction technique has been used. The computed diffracted far-fields of a typical HWF illuminated by a high performance primary feed shows good agreement with the available measured data and with the results based on the method of physical optics (PO). The sharp cutoff and the low spillover characteristics of the HWF are highlighted by comparing its radiation pattern with that of a hyperboloid without a flange. Further, the effects of the different parameters of the HWF on its radiation pattern are also studied and plotted, so that these results can be utilized in the design of the HWF for a specific requirement. Copyright © 1981 by The Institute of Electrical and Electronics Engineers, Inc.