HCR's Master Formula/Standard Formula-1 used in the Theory of Polygon can be rewritten in various forms of inverse trigonometric functions. 

The above Theorem was derived & proved by H C Rajpoot in July 2015. It applies to all regular and uniform polyhedra having vertex identical to that of a regular n-gonal right pyramid. 

The above Theorem of Rotation was derived & proved by H C Rajpoot in September 2019 at Indian Institute of Technology Delhi. 

The above corollary is a deduction from HCR's Theorem derived by H C Rajpoot

Description
A new term 'Circum-inscribed Polygon' has been introduced by the author in Geometry. A C-I polygon is a polygon which satisfies the conditions of a circumscribed polygon and an inscribed polygon together. In other words, the circum-inscribed polygon is a polygon which has both the inscribed and circumscribed circles. The above table shows the important and useful formula of a C-I polygon i.e. a C-I trapezium which is a trapezium having both inscribed and circumscribed circles.

Description
A regular n-gonal right antiprism is a semiregular convex polyhedron that has 2n identical vertices all lying on a sphere, 4n edges, and (2n+2) faces out of which 2 are congruent regular n-sided polygons, and 2n are congruent equilateral triangles such that all the faces have equal side. The above table shows the important parameters of a regular  

n-gonal right antiprism in terms of edge length a, and number of sides n in each regular polygonal face.

Description
This table presents the analytic formula to determine the different parameters of a regular pentagonal right antiprism in terms of side, such as normal distances of faces, normal height, the radius of a 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. 

Mathematical Proof
The maximum possible packing fraction of an infinite plane using identical circles of a finite radius is 

π√3/6≈ 90.69%.