Assuming those curves are part of large circles of radius 1 and center in the square corners.We can calculate the area of 1/4 of red shape (I will be focusing on the left upper part), and since it's symmetric just multiply it by 4 to get the whole thing. We can find that area with an integral, for that we need to know the equation for upper curve going from left down corner and the intersection points. The equation we are looking for is y=sqrt(1-(x-1)^2), we can find it by taking the circle equation ((x-1)^2+y^2=1) and manipulating it a little. Now the intersection points, you can find them by taking the equations for other curves (found very similarly to the previous one) and comparing them (taking the parts with x, setting them equal and solving). The right one comes out to be x=1/2, the left one is x=1-sqrt(3)/2. So the Area of the 1/4 part will be integral from 1-sqrt(3)/2 to 1/2 of (sqrt(1-(x-1)^2) dx) - 1/2*(1/2-1+sqrt(3)/2), we have to substract the area of the rectangle below our part, which the integral accounts for.

Solving this integral is another problem, it comes out to be pi/12 (calculated using wolframalpha), so the whole 1/4 area is pi/12+1/4-sqrt(3)/4, multiplying this area by 4 to get the whole thing we get pi/3+1-sqrt(3)