geometry - Formula to calculate a square root

I already know about the relation between a circle and the square root of a certain number. It looks like this
, where $b=\sqrt{x}$ and is actually very interesting. I wanted to check if I could take the image, and basically "write it down" in math form to come up with a, maybe useless, formula for the square root of $x$. The one I came up with is $$\sqrt{x}=\frac{(x+1)\sin(\arccos\big(\frac{x-1}{x+1}\big))}{2}$$ Now this formula is kinda long, and I want to know if, first of all, it is correct, and also if there is any possible way to simplify it without using a square root to achieve this, because that would be contradictory.