Hot rotating compound nuclei are modeled as ellipsoidally shaped liquid drops. A formalism for the determination of their equilibrium shape distribution is presented. Commonly used metrics associated with the volume element in deformation space are derived with a discussion of their underlying assumptions. Example calculations of shape distributions are presented. These distributions are not characterized by any one type of shape in particular; the full range of deformations from oblate to prolate are populated including spherical and triaxial deformations. However the tail of the distributions extending towards highly deformed prolate configurations is more pronounced especially at the higher angular momenta. The shape of the Coulomb-barrier distributions for the evaporation of charged particles from the nuclear surface, averaged over the equilibrium distribution of shapes, is calculated and its relevance for the understanding of the yield of low-energy a particles observed in evaporation spectra is discussed.