Perimeter of an Ellipse

Circumference of an ellipse: Exact series and approximate formulas.
Ramanujan I and Lindner formulas: The journey begins...
Ramanujan II: An awesome approximation from a mathematical genius.

Hudson's Formula and other Padé approximations.
Peano's Formula: The sum of two approximations with canceling errors.
The YNOT formula (Maertens, 2000. Tasdelen, 1959).
Euler's formula is the first step in an exact expansion.
Naive formula: p ( a + b ) features a -21.5% error for elongated ellipses.
Cantrell's Formula: A modern attempt with an overall accuracy of 83 ppm.
From Kepler to Muir. Lower bounds and other approximations.
Relative error cancellations in symmetrical approximative formulas.
Complementary convergences of two series. A nice foolproof algorithm.
Elliptic integrals & elliptic functions. Traditional symbols vs. computerese.
Padé approximants are used in a whole family of approximations...
Improving Ramanujan II over the whole range of eccentricities.
The Arctangent Function as a component of several approximate formulas.
Abed's formula uses a parametric exponent to fine-tune the approximation.
Zafary's formula. Improved looks for a brainchild of Shahram Zafary.
Rivera's formula gives the perimeter of an ellipse with 104 ppm accuracy.
Better accuracy from Cantrell, building on his own previous formula
Rediscovering a well-known exact expansion due to Euler (1773).
Exact expressions for the circumference of an ellipse: A summary.