Question 82
Suppose $A$, $B$, $C$, and $D$ are four circles of radius $r\gt 0$ centered about the points $(0, r)$, $(r, 0)$, $(0,-r)$, and $(-r, 0)$ in the plane. Let $O$ be a circle centered at $(0,0)$ with radius $2 r$. In terms of $r$, what is the area of the union of circles $A$, $B$, $C$, and $D$ subtracted by the area of circle $O$ that is not contained in the union of $A$, $B$, $C$, and $D$?
(The union of two or more regions in the plane is the set of points lying in at least one of the regions.)
Supplemental illustration (AI-generated):
(The union of two or more regions in the plane is the set of points lying in at least one of the regions.)
Supplemental illustration (AI-generated):