The question about the size of nuclei of formation of protofibrils (which constitute mature amyloid fibrils) formed by different proteins and peptides is yet open and debatable because of the absence of solid knowledge of underlying mechanisms of amyloid formation. In this work, a kinetic model of the process of formation of amyloid protofibrils is suggested, which allows calculation of the size of the nuclei using only kinetic data. In addition to the stage of primary nucleation, which is believed to be present in many protein aggregation processes, the given model includes both linear growth of protofibrils (proceeding only at the cost of attaching of monomers to the ends) and exponential growth of protofibrils at the cost of growth from the surface, branching, and fragmentation with the secondary nuclei. Theoretically, only the exponential growth is compatible with the existence of a pronounced lag-period (which can take much more time then the growth of aggregates themselves). The obtained analytical solution allows us to determine the size of the primary and secondary nuclei from the experimentally obtained concentration dependences of the time of growth and the new parameter-the ratio Lrel of the lag-time duration to the time of growth of amyloid protofibrils.