Actually I know how to calculate the circumference of an ellipse using two methods and each one of them giving me different result.

The first method is using the formula: $E\_c=2\pi\sqrt{\dfrac{a^2+b^2}{2}}$

The second method is determining the arc length of the first quart in the ellipse using **elliptic integral** multiplied by 4 (Look at picture below):

![Second method](https://i.stack.imgur.com/cApXM.jpg)

I want to know wath is the best method to get the exact circumference of an ellipse ?