Critique of circumference of ellipse

The formula for the circumference of an ellipse is not nearly so satisfying as it would be if we could find a closed form for the integral. We can't. Growing out of attempts to do so came the notion of elliptic integrals, and later elliptic functions, that involve expressions with square roots of cubic and quartic polynomials. The tools of elliptic function theory recently made headlines because they were used to prove Fermat's Last Theorem.

We conjectured that the circumference of an ellipse had the form (pi)(a + b). Even though the integral for the circumference doesn't simplify to this, a good question for further investigation is "How close are they?" That is, how good an approximation is (pi)(a + b) to the circumference of an ellipse with semimajor and semiminor axes a and b ?