Why is there no equation for the perimeter of an ellipse‽ - YouTube

Stand-Up Maths, by Matt Parker

Mentions several approximations.

For semi-axis lengths a, b, with a >= b:

Eccentricity:

Relative difference squared:

He mentioned these approximations:

Arithmetic mean:

Geometric mean:

Root mean square:

Srinivasa Ramanujan has two formulas:

(best linear + sqrt(quadratic))

(best of this form)

Matt Parker also mentioned some asymmetric formulas that have slightly erroneous small-eccentricity limits:

The exact formula is a complete Jacobi elliptic integral of the second kind

Using both argument conventions:

I used this page to test my formulas:

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