Perimeter of an Ellipse: Understanding Ramanujan's Approximation

TL;DR

This paper explains Ramanujan's highly accurate ellipse perimeter approximations and introduces improved formulas based on insights from the analysis, enhancing the understanding and accuracy of these classical formulas.

Contribution

The paper provides the first explanation of Ramanujan's ellipse perimeter formulas and proposes improved approximations with better accuracy.

Findings

Our approximation is uniformly more accurate than Ramanujan's.

We offer a novel derivation of Ramanujan's formulas.

The new formulas improve ellipse perimeter estimation.

Abstract

It is well known that there is no closed form analytic expression for the perimeter of an ellipse. In 1927, Srinivasa Ramanujan provides two approximations to the perimeter of an ellipse that are amazingly accurate. However, he does not provide an explanation of how he arrived at those expressions. In this paper, we will try to provide such an explanation that is likely how he derived those expressions. Using insights from our analysis, we improve on these approximations. To the best of our knowledge, ours is the first attempt to explain Ramanujan's ellipse perimeter formula and our approximation is uniformly better than his expression.