The discovery of a new family of high-T-C materials(1), the iron arsenides (FeAs), has led to a resurgence of interest in superconductivity. Several important traits of these materials are now apparent: for example, layers of iron tetrahedrally coordinated by arsenic are crucial structural ingredients. It is also now well established that the parent non-superconducting phases are itinerant magnets(2-5), and that superconductivity can be induced by either chemical substitution(6) or application of pressure(7), in sharp contrast to the cuprate family of materials. The structure and properties of chemically substituted samples are known to be intimately linked(8,9); however, remarkably little is known about this relationship when high pressure is used to induce superconductivity in undoped compounds. Here we show that the key structural features in BaFe2As2, namely suppression of the tetragonal-to-orthorhombic phase transition and reduction in the As-Fe-As bond angle and Fe-Fe distance, show the same behaviour under pressure as found in chemically substituted samples. Using experimentally derived structural data, we show that the electronic structure evolves similarly in both cases. These results suggest that modification of the Fermi surface by structural distortions is more important than charge doping for inducing superconductivity in BaFe2As2.