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The paper presents several possibilities for exact representation of the sphere obtained by composing rational multisided Bézier patches known as S-patches. We first derive a general method that utilizes the (inverse) stereographic projection and enables exact representation of a sphere section in terms of an S-patch over a regular polygon. Then, we apply the method to the faces of all five Platonic solids inscribed into the sphere. Depending on the Platonic solid, the obtained S-patches are defined over triangular, square or pentagonal domains and unify two previously known constructions based on triangular and tensor product Bézier patches. • Exact sphere representations based on rational multisided Bézier patches, also called S-patches, are provided. • The composite mapping method, which utilizes the stereographic projection, is used. • Representations of sphere parts over Platonic solids, inscribed into the sphere, are obtained. • The presented S-patches are defined over triangular, square or pentagonal domains. • Two previously known constructions, based on triangular and tensor product Bézier patches, are unified. [ABSTRACT FROM AUTHOR]