The rational approximation to the escape probability is generalized to contain a geometry dependent parameter. In this way, approximate expressions that are both simple and remarkably accurate are obtained for the escape probability from solid and hollow fuel rods, and for the Dancoff correction in regular rod lattices. These approximations are derived from suitably chosen one-parametric chord distribution functions that have the same general character as the exact chord distributions of the fuel and moderator regions. It is shown that it is reasonable to determine the parameter belonging to each geometry—the geometric index—from the condition that the logarithmic moment of the exact and the approximate chord distribution functions be equal. The geometric indices are given for solid and hollow fuel rods, and for square and hexagonal lattice configurations. For solid or hollow fuel rods the error in the approximation is less than 1 %. The Dancoff correction for rod lattices is obtained with comparable accuracy.