This paper studies the problem of controlling a multi-robot system to achieve a polygon formation in a self-organized manner. Different from the typical formation control strategies where robots are steered to satisfy the predefined control variables, such as pairwise distances, relative positions and bearings, the foremost idea of this paper is to achieve polygon formations by injecting control inputs randomly to a few robots (say, vertex robots) of the group, and the rest follow the simple principles of moving towards the midpoint of their two nearest neighbors in the ring graph without any external inputs. In our problem, a fleet of robots is initially distributed in the plane. The socalled vertex robots take the responsibility of determining the geometric shape of the entire formation and its overall size, while the others move so as to minimize the differences with two direct neighbors. In the first step, each vertex robot estimates the number of robots in its associated chain. Two types of control inputs that serve for the estimation are designed using the measurements from the latest and the last two time instants respectively. In the second step, the self-organized formation control law is proposed where only vertex robots receive external information. Comparisons between the two estimation strategies are carried out in terms of the convergence speed and robustness. The effectiveness of the whole control framework is further validated in both simulation and physical experiments.