The cut sphere fluid is studied in the isotropic phase by the Percus Yevick (PY) and the Hypernetted Chain (HNC) integral equation techniques, as well as by the theory recently proposed which is based on a geometrical interpretation of the direct correlation function. Fluids of cut spheres with thicknesses L* ranging from 0 to 0.7 have been studied, and detailed results for L*=0.1, 0.2, and 0.3 are reported. The L*=0 case is also examined. A new simplified version of the numerical implementation of the PY and HNC closures is proposed here. The results for pressures and structural properties are compared with the available simulations results and the recent theoretical results from the authors. The important feature of the present work is to show the ability of the HNC theory to predict the cubatic phase observed in the computer simulations for thicknesses around 0.2. The nematic phase is also predicted by the HNC theory for thicknesses smaller than L*=0.12. In agreement with previously obtained results, the detailed analysis of the PY theory results show that this approximation is unable to predict an instability toward any of the orientationally ordered fluid phases. The geometrical approach shows the correct trend for an isotropic to nematic transition, but exhibits an instability toward the cubatic phase only for thicknesses above L*=0.5, thus providing an illustration of the inability of standard density functional type theories to fully describe complex fluids. This study also sheds some light on the major differences between the three approaches in the treatment of many body density correlations.