We address the inverse problem of retrieving the shape of an obstacle with impedance in the form of a surface wave operator using the knowledge of electromagnetic scattering amplitude at a fixed frequency. We prove unique reconstructions from infinitely many measures. We then provide a characterization of the scattering amplitude derivative with respect to the obstacle shape. This derivative includes the case of shape dependent impedance parameters. We then employ a gradient-descent algorithm with H 1 boundary regularisation of the descent direction to numerically solve the inverse problem. The procedure is validated for three dimensional geometries using synthetic data.