What is the area of the blue region if $AB=120$.
I can only think in $\triangle OAB$ (O b being the low left point and the triangle is right) I can use Pythagoras which involves $120$, radius $r$ of the arc and diameter $d$ of the circle but I'm stuck there.
$$r^2=d^2+120^2$$
I know the area we are looking for is $$\frac{1}{4}\pi r^2 - \pi \left(\frac{d}{2}\right)^2$$
You were correct in your first equation. The diagram below shows this at a glance. Now we show that the area is the large quarter circle minus the small circle. \begin{align*} A&=\dfrac{\pi R^2}{4}-\pi r^2= \dfrac{\pi \big((2r)^2+120^2\big)}{4}-\pi r^2\\\\ &=\big(\pi r^2 + 3600\pi\big)-\pi r^2\\\\ &=3600\pi \end{align*}