I just stumbled upon
$$ \pi \approx \sqrt{ \frac{9}{5} } + \frac{9}{5} = 3.141640786 $$
which is $\delta = 0.0000481330$ different from $\pi$. Although this is a rather crude approximation I wonder if it has been every used in past times (historically). Note that the above might also be related to the golden ratio $\Phi = \frac{\sqrt 5 + 1}{2} $ somehow (the $\sqrt5$ is common in both).
$$ \Phi = \frac{5}{6} \left( \sqrt{ \frac{9}{5} } + \frac{9}{5} \right) - 1 $$
or
$$ \Phi \approx \frac{5}{6} \pi - 1 $$
I would like to know if someone (known) has used this, or something similar, in their work. Is it at all familiar to any of you?
Related Question (link).
Answer
Ramanujan found this approximation, among many others, according to Wolfram MathWorld equation 21 in linked page.
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