Application of HCR’s Theorem & Corollary Copyright©3-D Geometry by H. C. Rajpoot Mathematical Analysis of Pyramidal Flat Containers with Polygonal Base, Pyramids & Polyhedrons (Application of HCR’s Theorem and Corollary)     Master of Technology, IIT Delhi Introduction: In this paper, we are to derive the generalized formula to compute all the important parameters like V-cut angle (using HCR’s Theorem), edge length of open end, dihedral angle (using HCR’s Corollary), surface area and volume of pyramidal flat container with regular polygonal base, right pyramids and polyhedrons. We will apply these generalized formula to compute important parameters & make paper models of the pyramidal flat containers with square, regular pentagonal, hexagonal, heptagonal & octagonal bases, right pyramids and polyhedrons. The concept of making pyramidal flat containers, right pyramids and polyhedrons (bi-pyramids) is mainly based on the rotation or folding of two co-planar planes about their intersecting straight edges (as discussed in details and formulated in HCR’s Theorem) In brief, the procedure, of making pyramidal flat container, right pyramid or polyhedron using sheet (of desired thickness) of paper, polymer, metal or alloy which can be easily cut, bent & butt-joined at the mating lateral edges, is based on the following steps 1.) Making drawing on thin sheet of paper (we can also use sheet of polymer, metal or alloy) 2.) Cutting and removing undesired parts from sheet-drawing to get a blank 3.) Bending lateral faces (trapezoidal in case of flat container and triangular in case of pyramid) about the edges of regular polygonal base 4.) Joining the mating edges of lateral faces (gluing in case of paper, welding in case of polymer, metal or alloy) In above procedure, the most important step is to make drawing accurately & precisely on the sheet of desired material & thickness preferably sheet of paper is most suitable for making accurate drawing. We will discuss the procedure & the drawing in details in later stages. Pyramidal flat container with regular n-gonal base: Let’s make a right pyramidal flat container of slant height  (i.e. distance between parallel sides of a lateral trapezoidal face), with a regular polygonal base with  no. of sides each of length  such that each lateral trapezoidal face is inclined at an angle  with the plane of base. Let’s draw a circle of radius  circumscribing the regular n-gonal base           of each side . Draw a bigger concentric circle of radius  & extend the straight lines joining the vertices of inner regular n-gon to the centre O so that these lines intersect outer big circle at points            . After joining these points (vertices) we get another big regular n-gon           . (As shown in the fig-1) Now, interior angle () of regular n-gon,  (  )                    (   )     Drop a perpendicular from con-centre O to the parallel sides          . Figure-1: Inner and Outer circles of radii    circumscribe the inner regular n-gonal base & outer regular n-gon.      