This paper presents generalized analytical formulas for computing key geometric parameters of pyramidal flat containers with regular polygonal bases, right pyramids, and related polyhedra. The derived expressions include the V-cut angle (obtained using HCR’s Theorem), edge length of the open end, lateral edge length, dihedral angle (derived using HCR’s Corollary), surface area, and volume. These formulas provide a unified framework for the systematic analysis and construction of such structures for arbitrary regular polygonal bases. The results are applied to pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal, and octagonal bases, as well as to right pyramids and bipyramidal polyhedra. The underlying geometric construction is based on the rotation or folding of two coplanar planes about their intersecting straight edges, as formulated in HCR’s Theorem, and is illustrated through practical paper models.