Geometry

Let the intersection point directly above $P$ as $A$, and let the point that is to the left of $P$ be $B$

The area of $\triangle ABP = 12 \times 12 \div 2 = 72$ 

The area of the circular region $ABP$ is ${1 \over 4} \times 12^2 \times \pi = 36 \pi$

Now, note that the area of the shaded region is $2 (\text{circular region of ABP} - \triangle ABP)$

Substituting what we know, we find that the area of the shaded region is $\color{brown}\boxed{72 \pi - 144}$