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For planar near-field methodology, if the forward hemisphere is to be determined exactly, the propagating field must be sampled either in the aperture of the antenna or over a plane of infinite extent. In practice, due to the finite extent of the scan plane, any conventional planar near-field measurements inevitably represent a truncated data set, and, as such, any predicted far-field pattern includes errors associated with this truncation. We investigate the possibility of constructing bespoke polyhedral measurement surfaces that enclose the antenna under test and that are suitable for the derivation of the wide-angle antenna performance from measurements made using existing, possibly smaller, planar near-field measurement facilities. The paper shows, through numerical simulation and experimental measurement, that a flat-topped pyramid provides a possible solution to the measurement of high gain antenna patterns in the forward hemisphere using a planar scanner of size of order 1.5 times the size of the radiating aperture. Additionally, it is shown that by enclosing a medium gain antenna (e.g. a corrugated horn) within an imaginary box and measuring the near field on all six sides of the box, using a suitable rotation of the AUT, a prediction of the full spherical radiation pattern of the antenna can be obtained.