The Radial Product Framework: A Unified Method for Area and Volume Computation via Inradius-Exradius Duality

This paper introduces a family of geometric formulas termed the Radial Product Framework, which computes areas and volumes using inradius-exradius relationships. For triangles, the foundational formula ∆ = √ r · ra · rb · rc is derived, where r and ra,rb,rc denote the inradius and exradii. The framework is extended to bicentric quadrilaterals and tetrahedrons, with hybrid variants integrating angular and surface-area dependencies. These formulas bypass traditional reliance on side lengths or angles, offering novel computational pathways in both theoretical and applied geometry. Rigorous validation, examples, and applications are provided to demonstrate utility and originality.