The buckling properties of a carbon nanoring formed by bending a single-walled carbon nanotube and connecting the ends together are investigated by using molecular mechanics calculations. Armchair nanorings (8, 8) and zigzag nanorings (9, 0) with various ring diameters are used as examples. The calculations are conducted whilst the nanorings are subjected to displacement-controlled tension on the two symmetrical sides. The critical tension displacements are determined by calculating the potential energy of the nanorings. The buckling shapes of both armchair and zigzag nanorings with an odd number of units are unsymmetrical, whereas those with an even number of units are symmetrical.