Humans can exploit contacts anywhere on their body surface to manipulate large and heavy items, objects normally out of reach or multiple objects at once. However, such manipulation through contacts using the whole surface of the body remains extremely challenging to achieve on robots. This can be labelled as Whole-Body Contact-Rich Manipulation (WBCRM) problem. In addition to the high-dimensionality of the Contact-Rich Manipulation problem due to the combinatorics of contact modes, admitting contact creation anywhere on the body surface adds complexity, which hinders planning of manipulation within a reasonable time. We address this computational problem by formulating the contact and motion planning of planar WBCRM as hierarchical continuous optimization problems. To enable this formulation, we propose a novel continuous explicit representation of the robot surface, that we believe to be foundational for future research using continuous optimization for WBCRM. Our results demonstrate a significant improvement of convergence, planning time and feasibility - with, on the average, 99% less iterations and 96% reduction in time to find a solution over considered scenarios, without recourse to prone-to-failure trajectory refinement steps.