A recently developed approach, in the area of nonlinear oscillations, is used to analyze the single degree of freedom equation of motion of a oating unit (such as a ship) about a critical axis (such as roll). This method makes use of a closed form analytic solution, exact upto the rst order, and takes into account the the complete unperturbed (no damping or forcing) dynamics. Using this method very-large-amplitude nonlinear vessel motion in a random seaway can be analysed with techniques similar to those used to analyse nonlinear vessel motions in a regular (periodic) or random seaway. The practical result being that dynamic capsizing studies can be undertaken considering the shortterm irregularity of the design seaway. The capsize risk associated with operation in a given sea state can be evaluated during the design stage or when an operating area change is being considered. Moreover, this technique can also be used to guide physical model tests or computer simulation studies to focus on critical vessel and environmental conditions which may result in dangerously large motion amplitudes. Extensive comparitive results are included to demonstrate the practical usefulness of this approach. The results are in the form of solution orbits which lie in the stable or unstable manifolds and are then projected onto the phase plane.