Wikipedia says cycloid arc length is $8r$ and provides a straightforward integration to find it. But is it possible to find it with some elegant trick?
Doing it without using the calculus is somewhat complicated, and unfortunately difficult to explain without drawing pictures.
There is a very large literature on the problem. Begin your search by looking for "rectification of the cycloid." (Rectification is a largely obsolete term for finding arclength.)
Famous early rectifications of the cycloid are by Christopher Wren, by Roberval, and by Huygens. All of these rectifications come before the official birth of the calculus.