In this paper, we focus on recovering the impedance function or the boundary shape from a pair of Cauchy data on the known boundary by using an indirect boundary integral equation. This present problem has been divided into two parts. The first part is to solve a Cauchy problem through using an indirect boundary integral equation method combining a regularization technique. Then, the elastic impedance function is given by a point to point method. The second part is to recover the elastic shape by a Newton-type iterative method. The effectiveness of the method has been shown by solving some examples.