Value of the golden ratio (number Phi) knowing the side length of a square
DOI:
https://doi.org/10.17268/sel.mat.2021.02.16Keywords:
Number phi, Golden ratio, FibonacciAbstract
This paper explains how to obtain the number phi using a square with side length equal to a, the right triangle with sides a=2 and a, and a circle with radius equal to the hypotenuse of this right triangle. In particular, from a square whose side length is equal to a, we will show how to obtain a segment b in such a way that the value of a=b is the number phi. It is well known that this ratio is also calculated from equating the ratios obtained by dividing a segment of length a + b by a (being a always the largest segment) and a by b, that is, (a + b)=a = a=b. This equality is a consequence of the ratio of proportionality in triangles applying Thales’s Theorem. And, we must mention also how this golden ratio it is obtained as a consequence of the Fibonacci sequence. However, the golden ratio as a consequence of the limit of Fibonacci sequence was found later than many masterpieces, as for instance the ones of Leonardo da Vinci. This is the main reason because we analyzed how to find the proportionality golden ratio using the most common geometric figures and its symmetries. This paper aims to show how the golden ratio can be obtained knowing the side length a of a square.
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