This paper presents studies of the size effect on magnetic switching and interlayer magnetostatic coupling in synthetic antiferromagnet (SAF)-pinned spin-valve nanorings connected to nano stripes with a nanoconstriction. Micromagnetic simulation has been successfully used to reveal the detailed magnetization reversal process of these nanorings. It was observed that SAF-pinned spin-valve nanorings exhibit a double (onion-vortex-reverse onion) or single (onion-reverse onion) magnetization switching process depending on the ring diameter, which is contrary to narrow single-layer NiFe rings that show a double switching process only. Micromagnetic simulations of the SAF-pinned spin valve nanorings suggest that the magnetostatic coupling between the pinned layers of the SAF and the free layer is dependent on the ring size: it plays a very important role in the magnetization switching of small nanorings (600 nm outer diameter), but only a minor role in the switching of big nanorings (1600 nm outer diameter). An efficient reduction of the magnetostatic interaction results in a small shift of the minor magnetoresistance curve, which is beneficial to the magnetic memory applications.