Circle practice problem

 



Key point is to convince yourself that each vertex of the square is also the center of its nearby circular arc.
Once you do that you can see that area of the shaded region  = a^2 - 4*1/4*pi*(a/2)^2 = a^2(1 - pi/4).

Now how to convince yourself?
Notice that the arcs SR and RQ are touching each other at the point R which is the midpoint of CD.
So they share a common tangent passing through R.
Also SR and RQ arcs are mirror images of each other along the vertical line passing through R.
If they are mirror images of each other touching each other at R and the mirror is placed vertically at R then that mirror itself is the tangent.
If mirror is the tangent then the radius at that point will be perpendicular to it, and that will lie on the side CD.
Only point on CD which is equidistant from R and Q is C, hence that's the center. 

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