Deriving Ramanujan's Ellipse approximation

Ramanujan's ellipse approximation.

Taking the major and minor axises of an ellipse to be a and b, first you compute;

\[h = (a – b)/(a + b).\]

Then his approximation is the following.
\[\pi (a + b) \left(1 + \frac{3h^2}{10 + \sqrt{4 - 3h^2}}\right)\]

I can see how π•(a+b) can make sense as the closer a is to b the more circular the ellipse and the more accurate the approximation is as you basically have π•radius+radius or π•diameter. I do not understand how he got the rest of his equation. If anyone can show me how it's derived it'd be greatly appreciated!


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Erdos Erdos
  • Erdos Erdos

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