Knowledge-based protein folding potentials have proven successful in the recent years. Based on statistics of observed interatomic distances, they generally encode pairwise contact information. In this study we present a method that derives multi-body contact potentials from measurements of surface areas using coarse-grained protein models. The measurements are made using a newly implemented geometric construction: the arrangement of circles on a sphere. This construction allows the definition of residue covering areas which are used as parameters to build functions able to distinguish native structures from decoys. These functions, encoding up to 5-body contacts are evaluated on a reference set of 66 structures and its 45000 decoys, and also on the often used lattice ssfit set from the decoys'R us database. We show that the most relevant information for discrimination resides in 2- and 3-body contacts. The potentials we have obtained can be used for evaluation of putative structural models; they could also lead to different types of structure refinement techniques that use multi-body interactions.