AOB is a quadrant sector of radius r cut from a circle of center O. A semicircle of center P is drawn outside of quadrant taking AB as diameter. What is the difference between area of whole geometry and that of quadrant?
I want the area of ACBD (the shaded part). I proceeded as follows:
What next??
I agree with your calculations for the area of the quadrant sector and for the areas of the two parts in which the sector is cut by the chord $AB,$ namely, the triangle $\triangle AOB$ and the remaining part $ADBP,$ which is called a circular segment. I also agree about the radius of the semicircle.
The shaded region $ACBD$ is a type of figure called a lune.
From the radius of the semicircle you should be able to find its area, which is the combined areas of the circular segment $ADBP$ and the lune $ACBD.$ Then, knowing the area of the circular segment, you should be able to find the area of the lune easily.
This is actually a rather famous result in classical geometry.