This article is primarily addressed to the geometry of hard ellipsoidal molecules which is essentially required in the study of the fluid structure, thermodynamics and transport properties of the model hard-biaxial fluids or liquid crystals. A methodology has systematically been followed in obtaining the known as well as hitherto unknown results for a biaxial system. The computationally convenient and neat expressions have been achieved for the surface and volume elements of a biaxial molecule. New expressions have been obtained for the surface area S and the mean radius of curvature R of a rigid ellipsoid in terms of the elliptic integrals of the first and second kinds. In the Limit when the two axes become equal, the expressions obtained for the ellipsoidal molecules reduce to those of the spheroidal systems.,Exact expression has been obtained for the second virial coefficient of the ellipsoidal fluid and furthermore the significance of our analytical results for such a fluid are enunciated in the context of the study of the higher virial coefficients and equation of state.