When a wave of wavelength &lgr; is incident upon an opaque object of typical dimensiona, a shadow is formed in the geometric optics limit &lgr;/a=0. If &lgr;/ais small and not zero, the shadow boundary is shifted slightly from the geometrical shadow boundary as was first shown by Artmann. He found the shift to be asymptotic to &agr;(&lgr;2a)⅓for a circular cylinder, where &agr; is positive or negative according as the field or its normal derivative vanishes on the cylinder. The same result was obtained by Rice for a parabolic cylinder, but for the hard cylinder his &agr; differed from Artmann's. We have redetermined &agr; for the circular cylinder and found it to agree with the result for the parabolic cylinder in both cases. We have also determined the shift for a circular cylinder on which the field satisfies an impedance boundary condition. The former result is implicit in the work of Goriainov and both results are implicit in the work of Wait and Conda. We have also determined the scattering cross section of a circular cylinder with an impedance boundary condition. These results lead us to propose two formulas, one for the shift of the shadow boundary and one for the scattering cross section, of any smooth two‐ or three‐dimensional object. The latter expresses the deviation from the geometrical optics cross section as an integral, around a normal section of the shadow, of a multiple of the shift. This formula is verified for a sphere and for oblique incidence on a circular cylinder. Both electromagnetic and scalar waves are considered.