The Low Earth Orbit (LEO) satellite industry is undergoing rapid expansion, with operators competitively launching satellites due to the first-come, first-served principle governing orbital rights. This has led to the formation of increasingly large-scale, volumetric constellation where satellites operate across a diverse range of altitudes. To address the need for analyzing such complex networks, this paper establishes a new analytical framework for LEO constellations by leveraging a 3D Poisson point process (PPP). Specifically, we introduce a random height model (RHM) that can capture various altitude distributions by applying a random radial displacement to points generated by a homogeneous PPP on a nominal shell. Building on this, we derive an analytical expression for the downlink coverage probability. To motivate our model, we show that the altitude distributions of several leading satellite constellations, including Starlink, align with our model's assumptions. We then demonstrate through Monte Carlo simulations that the coverage probability of our RHM closely matches that of these real-world networks. Finally, we confirm the accuracy of our analytical expressions by showing their agreement with simulation results. Our work thereby provides a powerful tool for understanding and predict how the statistical distribution of satellite altitudes impacts network performance.