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Each petal of the region 
is the intersection of two circles, both of diameter 10. Each petal in turn is twice the area of a circular segment bounded by a chord of length 
, which implies the segment is subtended by an angle of 
. This means the area of the segment is
This means the area of one petal is
, and the area of is four times this, or 
.
Meanwhile, the area of
is simply the area of the square minus the area of , or 
.
So
(provided these regions are indeed disjoint; it's hard to tell from the picture)
This means the area of one petal is
Meanwhile, the area of
So