We examine the basic equilibrium states of two types structures, elastic lateral surfaces of truncated anticones and elastic Jordan curves on a sphere, depending on the angular excess of the lateral surfaces and the length of the curves, respectively. Experiments were performed to study the lateral surfaces, while numerical simulations were performed to study the Jordan curves. The latter were finally compared with experiments. We observed a good agreement between experiments and simulations in the case of free boundary conditions. Initially, the solutions have two-fold symmetry, but beyond a certain angular excess/length value, both types of structures prefer to occupy a configuration with skew symmetry.