Mathematically, a thin elastic rod is curve in the space with extra information about how the cross section rotates around it. Most older models refer to long, thin and zero curvature elastic isotropic rods with circular cross section. The authors focus on rods with non-circular cross section and arc-length dependent elasticity. After introducing the Kirchhoff equations for such an elastic rod, the authors consider an inverse problem approach for solving them. As part of the solution the unknown bending and torsional stiffness are obtained. It is interesting to note that the solutions obtained in this paper exhaust the class of solutions corresponding to elastic rods with constant curvature and torsion. The viability of the physical model is investigated.