Atomistic molecular dynamics simulations of a lipid bilayer were performed to calculate the free energy of a trans-membrane pore as a function of its radius. The free energy was calculated as a function of a reaction coordinate using a potential of mean constraint force. The pore radius was then calculated from the reaction coordinate using Monte Carlo particle insertions. The main characteristics of the free energy that comes out of the simulations are a quadratic shape for a radius less than about 0.3 nm, a linear shape for larger radii than this, and a rather abrupt change without local minima or maxima between the two regions. In the outer region, a line tension can be calculated, which is consistent with the experimentally measured values. Further, this line tension can be rationalized and understood in terms of the energetic cost for deforming a part of the lipid bilayer into a hydrophilic pore. The region with small radii can be described and understood in terms of statistical mechanics of density fluctuations. In the region of crossover between a quadratic and linear free energy there was some hysteresis associated with filling and evacuation of the pore with water. The metastable prepore state hypothesized to interpret the experiments was not observed in this region.