Fast prediction of the equilibrium configuration of a static floating rigid body is a challenging problem, especially if the body geometry is complex. To address this problem, a new model is developed to seek quickly the equilibrium configurations (position and orientation) of a continuous slender body with finite deformations relatively to the water surface. The non-linear description of the body geometry relies on the geometrically exact approach, where the orientations of the sections are parametrized by a continuous field of rotation matrices. After introducing the advantages of this approach, a new model giving the exact net wrench of buoyancy forces applied to the so-called Cosserat beam is reported. Then, an optimization method adapted to the Lie group manifold is developed to compute the equilibrium configurations. The numerical results are then compared to experimental data to validate the accuracy of the model. Finally, the inverse problem consisting of finding the relevant body deformation for a desired equilibrium configuration is introduced. After demonstrating that the inverse problem is ill-posed, a method is reported to deform continuously the body with a constant stable head configuration.