Golden Rhombus

A golden rhombus is a rhombus whose diagonals are in the ratio , where  is the golden ratio.

The faces of the acute golden rhombohedron, Bilinski dodecahedron, obtuse golden rhombohedron, rhombic hexecontahedron, and rhombic triacontahedron are golden rhombi.

The half-angle  is given by

			(1)
			(2)
			(3)
			(4)

(OEIS A195693).

Labeling the smaller interior angle as  and the larger as , then

(5)

and

			(6)
			(7)
			(8)
			(9)
			(10)
			(11)
			(12)
			(13)
			(14)
			(15)
			(16)
			(17)

(OEIS A105199 and A137218).

The diagonal lengths of a golden rhombus with edge length  are given by

			(18)
			(19)
			(20)
			(21)
			(22)
			(23)
			(24)
			(25)

(OEIS A121570 and A179290), the inradius by

(26)

and the area by

(27)

REFERENCES:

Kabai, S. Mathematical Graphics I: Lessons in Computer Graphics Using Mathematica. Püspökladány, Hungary: Uniconstant, pp. 177, 179, and 187, 2002.

Sloane, N. J. A. Sequences A105199, A121570, A137218, A179290, and A195693 in "The On-Line Encyclopedia of Integer Sequences."