AbstractWe validate a new law for the aspect ratio $\ensuremath{\alpha} = H/ L$ of vortices in a rotating, stratified flow, where $H$ and $L$ are the vertical half-height and horizontal length scale of the vortices. The aspect ratio depends not only on the Coriolis parameter $f$ and buoyancy (or Brunt–Väisälä) frequency $\bar {N} $ of the background flow, but also on the buoyancy frequency ${N}_{c} $ within the vortex and on the Rossby number $\mathit{Ro}$ of the vortex, such that $\ensuremath{\alpha} = f \mathop{ [\mathit{Ro}(1+ \mathit{Ro})/ ({ N}_{c}^{2} \ensuremath{-} {\bar {N} }^{2} )] }\nolimits ^{1/ 2} $. This law for $\ensuremath{\alpha} $ is obeyed precisely by the exact equilibrium solution of the inviscid Boussinesq equations that we show to be a useful model of our laboratory vortices. The law is valid for both cyclones and anticyclones. Our anticyclones are generated by injecting fluid into a rotating tank filled with linearly stratified salt water. In one set of experiments, the vortices viscously decay while obeying our law for $\ensuremath{\alpha} $, which decreases over time. In a second set of experiments, the vortices are sustained by a slow continuous injection. They evolve more slowly and have larger $| \mathit{Ro}| $ while still obeying our law for $\ensuremath{\alpha} $. The law for $\ensuremath{\alpha} $ is not only validated by our experiments, but is also shown to be consistent with observations of the aspect ratios of Atlantic meddies and Jupiter’s Great Red Spot and Oval BA. The relationship for $\ensuremath{\alpha} $ is derived and examined numerically in a companion paper by Hassanzadeh, Marcus & Le Gal (J. Fluid Mech., vol. 706, 2012, pp. 46–57).