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Above, I've highlighted several regions which we will use as intermediate steps to find the area of the center. The letters A through D represent the area of that region. I'm taking the square to have unit length sides.
A is a sixth of a circle.
A = 1/6 pi
B is an equilateral triangle.
B = sqrt(3)/4
C can be found by combining A and B.
C = 2A - B = 1/3 pi - sqrt(3)/4
D is a quarter of a circle.
D = 1/4 pi
By tinkering around with C and D, we can find the area of the center region.
Area of center = 1 - 4D + 4C = 1/3 pi + 1 - sqrt(3)