The circular ensembles of Dyson satisfy isoperimetric inequalities and concentration of measure phenomena for large particle numbers analogous to the isoperimetric inequality for surface measure on the sphere in Euclidean space of high dimension. This leads to a geometrical proof of a result of Johansson [Bull. Sci. Math. (2) 112, (1988), 257-304] that the empirical distribution of energy levels under such ensembles converge weakly almost surely to normalized arclength on the unit circle as n→∞.