I wanted to compare answer for area of the region in the middle of the square, the leaf sort of shape. So the circle is like a wheel and as it turns to the other side, it basically draws the upper half of the leaf and same process with the lower half.
From this the square would have side length 2 and width $\pi$ because the distance covered is half of the circumference.
I apologise for any mis-comunication as this is supposed to be an animation however I only have a screen shot.
For a visual answer if you look at the distance between he two curves. IT matches up perfectly with the equal distance of the height of the circle as is demonstrated here:
https://www.youtube.com/watch?v=0iUpZ3FfyH0