Derivation of the radius of gyration and aspect ratio of a circular, constant thickness patch on a spherical surface

In this technical note, we present a direct derivation of closed-form expressions for the following parameters for a circular, constant thickness patch on a spherical substrate, where the patch thickness and coverage can be varied independently: The moment of inertia of a patch about its symmetry axis. The radius of gyration of the patch. The aspect ratio of the patch, defined as the ratio of an effective “diameter” (twice the radius of gyration) to the patch thickness. The relationship between patch volume, coverage, and aspect ratio. These analytical expressions not only fill a crucial gap in the technical literature but also provide a quantitative basis for assessing how variations in patch morphology influence mechanical, dynamic, and optical properties, thereby offering valuable guidance for both theoretical analysis and experimental design.