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

*a* and semiminor axis

*b* 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

*a* and

*b*.

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.