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Geometry |
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Authors: Harish Chandra Rajpoot
In this paper, a comprehensive mathematical analysis of the rhombic dodecahedron is presented, and closed-form analytical expressions are derived for a polyhedron consisting of 12 congruent rhombic faces, 24 edges, and 14 vertices lying on a common circumscribed sphere. Using HCR’s Theory of Polygon, generalized formulae are obtained for the face angles and diagonals of the rhombic faces, as well as for the radii of the circumscribed sphere, inscribed sphere, and midsphere. Analytical expressions are further derived for the total surface area and enclosed volume in terms of the edge length. In addition, the solid angles subtended at the vertices and the dihedral angles between adjacent faces are evaluated. It is also shown that this convex polyhedron can be constructed by assembling twelve congruent right pyramids with rhombic bases and a specific normal height.
Comments: 11 Pages, 9 Figures
Download: PDF
[v1] 2026-02-05 23:11:58
Unique-IP document downloads: 94 times
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