On the Perimeter of an Ellipse

Computing accurate approximations to the perimeter of an ellipse is a favorite problem of mathematicians, attracting luminaries such as Ramanujan [1, 2, 3]. As is well known, the perimeter of an ellipse with semimajor axis and semiminor axis can be expressed exactly as a complete elliptic integral of the second kind.What is less well known is that the various exact forms attributed to Maclaurin, Gauss-Kummer, and Euler are related via quadratic hypergeometric transformations. These transformations lead to additional identities, including a particularly elegant formula symmetric in and .Approximate formulas can, of course, be obtained by truncating the series representations of exact formulas. For example, Kepler used the geometric mean, , as a lower bound for the perimeter. In this article, we examine the properties of a number of approximate formulas, using series methods, polynomial interpolation, rational polynomial approximants, and minimax methods.