The catalytic activities of the Fe-N-C single-atom catalysts (SACs) are associated with the varying atomic interactions through its characteristic coordination geometry. Yet, modulation of the surface curvature of carbon acting as a supporting body has not been investigated. Herein, we report the superior catalytic activity for the oxygen reduction reaction (ORR) and enhanced performance for urea oxidation reaction (UOR) of single Fe atoms anchored on a highly curved N-doped carbon dodecahedron with concave morphology (Fe SA/NhcC). Theoretical calculations and in situ spectroscopy disclose that the curvature of the carbon support helps to shorten the bond length of Fe-N, spatially redistributing the charges around the Fe and thereby lowering the d-band center toward optimal adsorption for oxygenated species. The Fe SA/NhcC catalyst displays an ultrahigh half-wave potential of 0.926 V for ORR and a small potential difference of 0.686 V for bifunctional ORR/UOR. A rechargeable Zn-urea-air battery with the Fe SA/NhcC cathode displays robust discharge durability, excellent cycling lifespan and higher energy efficiency compared to conventional Zn-air batteries. This work provides new insight into promoting the catalytic activity of SACs through varying the surface curvature of the supporting carbon, tailoring geometric configuration and electronic states of SACs.