Geometric and probabilistic structure of the uniform distribution on the sphere S2

This paper presents a self-contained exposition of the geometric and probabilistic structure of the uniform distribution on the unit sphere S2⊂R3. Relying only on spherical coordinates, symmetry principles and basic calculus, we derive fundamental properties of the uniform measure including the distributions of latitude, geodesic angles, spherical distances, Euclidean distances, spherical caps and dot products. The paper further develops explicit formulas for expectations of rotationally symmetric functions and provides closed-form expressions for mean geodesic quantities that arise naturally in geometric probability and quantization theory. In addition, we explain how rotational invariance simplifies spherical integration and leads to transparent interpretations of one-mean and multi-mean geodesic quantization on S2. The presentation is designed to be accessible to students and researchers seeking an elementary yet rigorous introduction to spherical probability laying a foundation for further study in geometric analysis, directional statistics, and quantization on curved surfaces.