Public school bus routes can change from year to year as students and their home locations change. However, school administrators benefit from the ability to predict future transportation needs on multi-year time scales. With this motivation in mind, this paper develops planning models for school bus routing when student locations are not known with certainty. Continuous approximation methods have been shown to be valuable in other settings where future spatial realizations of transportation demand are unknown. We first show how applying standard continuous approximation techniques to school bus routing entails unique methodological challenges not found in other routing problems. Specifically, we argue that the typical square-root functional form derived from the classical Beardwood-Halton-Hammersley Theorem does not necessarily produce appropriate route length approximations for a broad class of planar covering route problems, in which vehicles visit facilities (e.g., school bus stops, parcel lockers) that collectively ‘cover’ demand points (e.g., students’ homes, delivery recipients) within a certain radius. We provide both empirical evidence and analytical results in support of this argument. Next, we propose a continuous approximation framework for robust estimation of a school’s minimum required bus fleet size when student locations are distributed independently and identically at random. We focus on the non-trivial case in which the maximum allowable route duration (and not the physical bus capacity) is the limiting factor. Prior experience with school administrators suggests that the ability to quickly evaluate ‘what-if’ scenarios in real time is critical during the planning process; therefore, our approach requires minimal computation after one-time setup. Finally, we conduct computational experiments to validate our approach for a stylized planar region and actual school attendance regions in the United States.