(Note: e is the eccentricity from above)

But calculating it needs an infinite amount of terms ("Infinite Series 1" above).

Comparing

Just for fun, I calculate the perimeter using the three approximation formulas, and the two exact formulas (but only the first four terms, including the "1", so it is still just an approximation) for selected values of a and b:

		Circle				Lines
a:		10	10	10	10	10
b:		10	5	3	1	0
Approx 1:		62.832	49.673	46.385	44.65	44.429
Approx 2:		62.832	48.442	43.857	40.606	39.834
Approx 3:		62.832	48.442	43.859	40.639	39.984
Series 1:		62.832	48.876	45.174	43.204	42.951
Series 2:		62.832	48.442	43.859	40.623	39.884
Exact*:		20π				40

* Exact:

• When a=b, the ellipse is a circle, and the perimeter is 2πa (62.832... in our example)
• When b=0 (not really an elllipse, just two lines back and forth) the perimeter is 4a (40 in our example)

They all get the perimeter of the circle correct, but only Approx 2 and 3 and Series 2 get close to the value of 40 for the extreme case of b=0.

Ellipse Perimeter Calculations Tool

This tool does the calculations from above, but with more terms for the Series.