Why Doesn't the Arc Length Formula of the Cycloid have π in it?

Question

So basically what I was thinking is if a cycloid curve is made by a rolling circle then its length should include $\pi$ somehow. I understand it's not the same length as the circle itself ($2\pi r$), but how can $π$ disappear from the formula, namely $8r$?

While on the other hand the area under one arch of a cycloid is given by $3\pi r^2$, $3$ times the area of the rolling circle, so still includes the $\pi$.

Finally I have a small hypothesis ,that this happened because the actual distance covered by the circle includes $\pi$ (which is $2\pi r$) so maybe they cancel each other somehow but I have no clue how to show that.