Mathematical analysis of regular pentagonal right antiprism

A regular pentagonal right antiprism is a convex polyhedron that has 10 identical vertices all lying on a sphere, 20 edges, and 12 faces out of which 2 are congruent regular pentagons, and 10 are congruent equilateral triangles such that all the faces have equal side. This paper presents, in detail, the mathematical derivations of the analytic formula to determine the different parameters in terms of side, such as normal distances of faces, normal height, the radius of the circumscribed sphere, surface area, volume, dihedral angles between adjacent faces, and the solid angle subtended by each face at the center, using the known results of a regular icosahedron. All the analytic formulae have been derived using simple trigonometry, and 2-D geometry which are difficult to derive using any other methods. A paper model of a regular pentagonal right antiprism with an edge length of 4 cm has been made by folding the net of faces made from an A4 white sheet of paper.