As you all know, when we inscribe a circle (with radius 1) into a square with side length 2, we cannot approximate the circle's circumference by "folding in" the corners of the square because the circumference of the square will remain 8 and will never be $2 \cdot \pi$. The error doesn't get smaller regardless of how often we fold in the corners.
We can, however, approximate the circle's area by "folding in" the corners of the square. But I'm stuck at the calculation.
What do I mean by "folding in"? In the first picture at https://i.stack.imgur.com/nOl6h.jpg, you can see a quarter circle with center (1,1) and radius 1 inscribed in a square with vertices (0,0), (2,0), (2,2) and (0,2). To approximate the area of the circle, I first fold in the bottom-left corner such that the new vertex is on the circle and such that the line segment between the origin and the new vertex is the diagonal of a square. As a result, a square (with vertices A', B', C' and D' in the picture) is subtracted (see picture 2). Subtracting this square 4 times (at each corner of the original square) completes Step 1.
Now repeat these steps at each vertex that is not touching the circle (so at 8 vertices after Step 1).
The more often you repeat these steps, the better the remaining area approximates the area of the circle.
Using some geometry, I was able to figure out that the length of the line segment between A' and B' must be $\frac{\sqrt{2}-1}{2} \approx 0.29289321881$ and that the length of the line segment between $A_1$'' and $B_1$'' must be $\frac{1}{2}\cdot (\frac{\sqrt{2}+1}{2}-\sqrt{\frac{2\sqrt{2}+1}{2}}) \approx 0.16177785576$, which coincides with what GeoGebra shows when constructing all of this. I didn't calculate the length of the line segment between $A_1$''' and $B_1$''', but according to GeoGebra it is approximately 0.1030985186 and the length between $A_3$''' and $B_3$''' is approximately 0.0686032506, so together the two squares added in step 3 have an area of approximately 0.01533571 square units.
As you can see though, the side lengths of these squares seem to get more and more complicated very quickly. I was wondering if there is a sequence for the side lengths of these ever shrinking squares but can't seem the see the pattern. Because then, with 4 square units being the initial area, I could calculate $\pi$ as: 4 - (4 $\cdot$ area of square in step 1 + 8 $\cdot$ area of square in step 2 + 8 $\cdot$ area of the two squares in step 3 + ... ), which after the first 3 steps comes out as approximately $3.324791968$.
Thank you in advance for any advice!