Stable structures of a rolled seamless belt of atoms arranged in a two-dimensional honeycomb lattice such as a carbon nano-belt, transit from a ring-shaped structure with a rotational invariance to folded structures as the circumference of the belt increases. Using molecular mechanics, we consider the size dependence of the structures obtained by simulated annealing. In addition, the relationship between the variety of metastable structures and the rate of temperature decrease during annealing is also examined. Furthermore, we investigate the effects of the van der Waals interaction between the surfaces on the structural transitions.