Calculations of the pressure distention of closed cylindrical vessels using classical infinitesimal-strain theory predict that, for isotropic materials, the length remains fixed while the diameter increases linearly with pressure. These predictions can be verified experimentally only if the radial deformation is less than 2–3%. This paper develops formulae applicable to deformations up to approximately 10 times the above, based on a modification of infinitesimal theory. The results predict significant lengthening of isotropic vessels, and ballooning or ‘blow-out’ above a certain pressure. It is shown that the classical stresses contain a common hydrostatic component which must be subtracted before the stresses can be integrated across the wall thickness to yield wall tensions. When the hydrostatic component is taken into account, Laplace’s law is found to hold for thick-walled vessels as well as for thin-walled vessels.