Background: Equations of state for a cold neutron star's interior are presented in three-column tables that relate the baryonic density, the energy density, and the pressure. A few analytical expressions for those tables have been established these past two decades as a convenient way to present a large number of nuclear models for neutron-star matter. Some of those analytical representations are based on nonunified equations of state, in the sense that the high- and the low-density part of the star are not computed with the same nuclear model. Purpose: Fits of equations of state based on a piecewise polytropic representation are revised by using unified tables of equations of state, that is to say, models which have been calculated consistently for the core and the crust. Methods: A set of 52 unified equations of state is chosen. Each one is divided in seven polytropes via an adaptive segmentation, and two parameters per polytrope are fit to the tabulated equation of state. The total mass, radius, tidal deformability, and moment of inertia of neutron stars are modeled from the fits and compared with the quantities calculated from the original tables to ensure the accuracy of the fits on macroscopic parameters. Results: We provide the polytropes parameters for 15 nucleonic relativistic mean-field models, seven hyperonic relativistic mean-field models, five hybrid relativistic mean-field models, 24 nucleonic Skyrme models, and one ab initio model. Conclusions: The fit error on the macroscopic parameters of neutron stars is small and well within the estimated measurement accuracy from current and next-generation telescopes.