How to determine the arc length of ellipse? I want to determine the length of an arc from the ellipse in the picture below:

How can I determine the length of $d$?

I believe I found this asymptotic approximation from Jacobi's elliptical integral of the second kind some time ago. It's not precise, but converges exactly for degenerate cases of $b=0$ (lines) and $b=a$ (circles). The infinite series methods are ideal where precision is desired. I offer this only as a curiosity. The last term, recently added, contributes little, but ensures convergence for circles. Assume $0\le b\le a$. Then
$$
s\approx 2 \pi \sqrt{a b}+(4 a)^{\frac{a-b}{a+b}} -a^{b-a}.
$$
