Generalized Analytical Geometry and Paper Models of Pyramidal and Polyhedral Forms

In this work, the closed-form analytical expressions were developed for determining the significant geometric parameters of pyramidal flat containers with regular polygonal bases, regular right pyramids, and associated polyhedral forms. The formulation is based on HCR’s Theorem of Rotation and its corollary, which provide explicit relations for the V-cut angle required for folding, the edge length of the open polygonal end, the lateral edge length, the dihedral angles between adjacent faces, and the corresponding surface area and enclosed volume. These results establish a unified analytical framework for the systematic analysis and constructive design of such structures for arbitrary regular n-gonal bases. The applicability of the derived relations is demonstrated through representative configurations of pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal, and octagonal bases, and is further extended to regular right pyramids and bipyramidal polyhedra formed by joining two identical pyramidal units. The underlying geometric construction is interpreted in terms of the controlled rotation or folding of initially coplanar planes about their intersecting straight edges, and its practical realization is illustrated through simple paper models.