In a simply connected planar domain D, a pair of Cauchy data of a harmonic function u in D is given on an accessible part of the boundary curve. On the non-accessible part, u is supposed to satisfy a homogeneous impedance boundary condition. We consider the inverse problem to recover the non-accessible part of the boundary or the impedance function. Our approach is based on a method that has been suggested by Kress and Rundell (2005) for recovering the interior boundary curve of a doubly connected planar domain. For comparison we give a short review of a method based on a single-layer potential approach suggested by Cakoni and Kress (2007) for solving the inverse problem. The integral equations obtained from both approaches can also be used for the problem to extend the incomplete Cauchy data and to solve the inverse problem to recover the impedance profile on a known boundary curve. For the latter problem we give numerical examples to show the feasibility of the approaches. Since in practice corrosion can lead to corners on the boundary we also incorporated them in the numerical treatment.