Mathematical analysis of truncated hexahedron (cube) AppliĐatioŶ of HCR’s forŵula for regular polyhedroŶs ;all five platoŶiĐ solidsͿ Applications of HCR’s Theory of PolygoŶ proposed ďy Mr H.C. Rajpoot (year-2014) ©All rights reserved                    Mr Harish Chandra Rajpoot M.M.M. University of Technology, Gorakhpur-273010 (UP), India Dec, 2014 Introduction: A truncated hexahedron (cube) is a solid which has 8 congruent equilateral triangular & 6 congruent regular octagonal faces each having equal edge length. It is obtained by truncating a regular hexahedron (having 6 congruent faces each as a square) at the vertices to generate 8 equilateral triangular & 6 regular octagonal faces of equal edge length. For calculating all the parameters of a truncated hexahedron, we would use the equations of right pyramid & regular hexahedron (cube). When a regular hexahedron is truncated at the vertex, a right pyramid, with base as an equilateral triangle & certain normal height, is obtained. Since, a regular hexahedron has 8 vertices hence we obtain 8 truncated off congruent right pyramids each with an equilateral triangular base. Truncation of a regular hexahedron (cube): For ease of calculations, let there be a regular hexahedron (cube) with edge length   & its centre at the point C. Now it is truncated at all 8 vertices to obtain a truncated hexahedron. Thus each of the congruent square faces with edge length  is changed into a regular octagonal face with edge length   (see figure 2) & we obtain 8 truncated off congruent right pyramids with base as an equilateral triangle corresponding to 8 vertices of the parent solid. (See figure 1 which shows the truncation of a regular hexahedron (cube) & a right pyramid with equilateral triangular base of side  & normal height  being truncated from the regular hexahedron).                                                   Figure 1: A right pyramid with base as an equilateral triangle with side length  & normal height h is truncated off from a regular hexahedron (cube) with edge length  +   Figure 2: Each of the congruent square faces with edge length  +   of a regular hexahedron is changed into a regular octagonal face with edge length  by truncation of vertices.