ABSTRACT We study the structure of contact binaries assuming a polytropic relation between pressure and density, restricting ourselves to the case of equal-mass components, i.e. symmetric contact binaries. In this case, matter is at rest in the corotating reference frame making this problem far simpler than the general case of non-symmetric contact binaries. We compute these structures assuming values of the polytropic index of n = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, and 3.5 employing a self-consistent technique due to Hachisu. As a part of the results, we find the shape of their surfaces. While for the case of n = 3.5, such surfaces are very close to those corresponding to equipotentials of the restricted Lagrangian three-body problem, for lower n values the departure is remarkable. We propose a generalized function to fit these surfaces, which allows us to perform an accurate integration of the light curve due to the object. Then, for values of n > 0.0 we computed a family of light curves considering different inclinations and values for the width of the neck connecting the components (or equivalently, the fillout factor) of the pair. We compare our calculations with the solution found for the symmetric contact binary V803 Aquilae by employing the popular phoebe code, that assumes the above-mentioned Lagrangian equipotential surfaces. We conclude that considering polytropic structures leads to parameters that may be appreciably different from those deduced by assuming that their surfaces correspond to equipotentials of the restricted three-body problem.