An Exact Connection between π and the Golden Ratio φ

  •  Luis Teia    


This article establishes a geometrical and mathematical bridge between the universal constant π  and the golden ratio φ  by interrelating the construction of the area of a circle using two approaches: one formed using the rotation of a regular unit pentagon and the other one from the rotation of its inverse ― the pentagram of side reference φ . The mathematical end result is a linear expression of π  as a function of φ . As an interesting side result, an expression for the area of a circle is derived based on the golden ratio φ  and a geometrically motivated coefficient. A scripted program for the verification of the derived expressions is provided.

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