We study a mathematical model for induction hardening of steel. It consists of a vector potential formulation of Maxwell's equations coupled with a heat equation and an evolution equation for the volume fraction of the high temperature phase in steel called austenite. An important task for practical applications of induction hardening is to find the optimal coupling distance between inductor and workpiece. To this end we control the volume fraction of austenite with respect to perturbations of the coupling distance. The coil is modeled as a tube and is defined by a regular curve. We formulate the shape optimization problem over the set of admissible curves and prove the existence of an optimal curve. We apply the material derivative method for the shape sensitivity analysis of the state system. Finally, the shape gradient is specified for an optimal curve and the first order necessary optimality conditions are established.