@peterstucklschwaiger2830

It's even simpler, I'd say. 

Rotate the smaller part of the “claw” (AYS - as you said). 
Calculate the r=6 quarter circle (AYS): 9pi
Subtract 3 whole s=2 squares (AY partly S): 9pi - 12
Subtract 2 s=2 half squares : 9pi - 16
Subtract the r=2 quarter circle: 8(pi - 2)