The structures of the 6H perovskites Ba3
B
2+Sb5+
2O9, B = Ca and Sr, have been solved and refined using synchrotron X-ray and neutron powder diffraction data. Ba3CaSb2O9 and Ba3SrSb2O9 have monoclinic C2/c and triclinic P\bar 1 space-group symmetries, respectively, while Ba3MgSb2O9 has ideal hexagonal P63/mmc space-group symmetry. The symmetry-lowering distortions are a consequence of internal `chemical pressure' owing to the increasing effective ionic radius of the alkaline-earth cation in the perovskite B site from Mg2+ (0.72 Å) to Ca2+ (1.00 Å) to Sr2+ (1.18 Å). Increasing the effective ionic radius further to Ba2+ (1.35 Å) leads to decomposition at room temperature. The driving force behind the transition from P63/mmc to C2/c is the need to alleviate underbonding of Ba2+ cations in the perovskite A site via octahedral rotations, while the transition from C2/c to P\bar 1 is driven by the need to regularize the shape of the Sb2O9 face-sharing octahedral dimers. Ab initio geometry-optimization calculations were used to find a triclinic starting model for Ba3SrSb2O9.