Can anyone answer this sat math question?

From my diagram,(Fig 1)

As PQRS is a square,

PQ=QR=RS=SP=a

Also,PQ=BC=a ; where BC is the diameter of the circle

ED , which is the diagonal of the inner square is also the diameter of the circle.

So,BC=ED=a

Thus,Radius of the circle,r=a/2

Now,from Fig 2 ,

In square HDGE,

∆DGE,is a right-angled triangle.

Applying Pythagoras theorem for right-angled triangle,

DG^2+EG^2=ED^2

We can get the side of the inner square.

DG^2+DG^2=ED^2

or,2DG^2=a^2

or,DG^2=(a^2)/2

or,DG=a/√2

Now,Area of the shaded region,

=Area of the circle-Area of the inner square

=π*r*r-(side)*(side)

=π(a/2)^2 - (a/√2)*(a/√2)

=π(a^2)/4 - (a^2)/2

=(a^2)*((π/4)-(1/2))

=(a^2)*((π-2)/4)

[ANS: (A)]