Explore modern developments in polygon area formulas and polyhedra volumes in this 34-minute mathematics lecture. Delve into the work of Moebius, Bowman, and Robbins on cyclic pentagon areas. Investigate 3D generalizations of Heron's formula and Archimedes' theorem for polyhedra, tracing back to Tartaglia's tetrahedron case. Examine connections to Cauchy's rigidity theorem, R. Connelly's example, and the Bellows Conjecture resolved by I. Sabitov. Discover how quadrances, rather than side lengths, form the basis of these formulas, highlighting the intrinsic algebraic nature of this geometry. Cover topics including cyclic quadrilateral and pentagon formulas, Tartaglia's and Euler's contributions, flexible polyhedra, and octahedra.