But can't you just use integration on the arc length?

 I recently came across the following video 

It made me think surly this can be derived in the same way as one can derive the formula for a circle. I'd gone through that derivation a number of times at school and since the formula for an ellipse is so close thought the same approach would surly work for an ellipse.

I decided to give it a go believing that I surely wouldn't get the answer I expected but it would be interesting to see why the arc-length method wouldn't work.

At this point, if I had to guess I would guess that the impossibility lay in the issue with integration.

The formula for a circle can be represented parametrically as:

To adapt this to an ellipse one simply needs to add constants in front of each trigonometric function:

The arc length of a line can be calculated by the following identity which follows from Pythagoras:

Using this identity we can get a formula for s in terms of x and y. However, this is where the signs of the problem begin as to find s from ds we need to perform an integration.

After thinking a little about how I might attempt to integrate this I try wolfram alpha to see where this might end up. Partially expecting it to say that there is no definite representation of the integral and that instead a numerical method would be required.

To my surprise, it gives a solution using a function I hadn't heard of before!

What is this elliptic integral of the second kind?

It turns out that rather than accept that the integral couldn't be done mathematics supposed that it could be done and then finds ways to join lots of different integrals together to use this common function.

This is not dissimilar to the log product which I discovered when trying to solve equations in wolfram alpha which I didn't previously understand how to solve.

The log product supposed that there exists a solution to the following equation, calling the solution W.

That isn't dissimilar to how the natural logarithm function works.