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 $d$?

You can compute this as
$$d=b\,E\bigl(\tan^{-1}(a/b\,\tan(\theta))\,\big|\,1-(a/b)^2\bigr)$$
using the incomplete elliptic integral of the second kind $E(\varphi\,|\,m)$. In Mathematica-Syntax (and suitable for Wolfram Alpha) this can be written as
2.23*EllipticE[ArcTan[3.05/2.23*Tan[50°]],1-(3.05/2.23)^2]

I adapted this from this post which investigates the converse problem (given arc length, find angle) but along the way treats this direction of the problem as well. As noted there, this angle conversion will only work for the first and last quadrant. Otherwise, either adjust the angle or look at that post for an alternative formula to use in its place.
With a few more digits of precision, the answer is returned as $2.5314195265536624417$ which essentially matches both the other answers here. Of course, printing that many digits in the answer is very bad style if the input is only given to two decimals. It does show that the numerical integration by Jyrki is a bit less precise than what coffeemath did, but even he should theoretically have rounded in the other direction.
Note that the formula above only works for $-\frac\pi2<\theta<\frac\pi2$. For $\frac\pi2<\theta<\pi$ the result of $\tan^{-1}$ will represent $\theta-\pi$ so to correct for that you can add $\pi$ to that result. Similar for $-\pi<\theta<-\frac\pi2$ where you need to subtract $\pi$ from the $\tan^{-1}$ result. Generally speaking you want the first input to the $E$ function to be an angle in the same quadrant as $\theta$, adding integer multiples of $\pi$ as necessary.
