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Two circles share a common chord $\overline{PQ}.$ The chord cuts off a $90^\circ$ arc of circle $A$ and a $60^\circ$ arc of circle $B$. If the area of quadrilateral $APBQ$ is $6+6\sqrt 3.$ Find the combined area of the gray regions.

 Feb 20, 2021
 #1
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The area of the gray region is 26*pi.

 Feb 20, 2021
 #2
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PQ / 2  ==>  x

 

x*( x + x / tan30º ) = 6 + 6√3        x = 2.449491

 

Radius  AP = x / sin45º = 2√3

 

Radius  BP = x / sin30º = 4.898979486

 

Area = [AP2*pi (270/360)] + [BP2*pi (300/360)]

 

Area = 29pi

 Feb 21, 2021

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