A square inscribed in a unit circle has diagonal length 2 and its area (2) forms a lower bound for pi.
A square circumscribed around a unit circle has side length 2 and its area (4) forms an upper bound for pi.
I’m imaging rotating and scaling the inscribed square so that the length ‘across’ it remains 2 but this goes from being the inscribed to circumscribed, from 2 as the diagonal to 2 as the width.
This rotation would create 4 small areas inside and 4 outside the circle. 4 Outside areas could be thought of as a kite minus a sector, 4 inside areas could be thought of as a sector minus a triangle. When the kite-sector equals the sector-triangle the area of the square would be pi. (?!).
The square that makes this happen is the square equal to the circle. I’m stuck on the algebra of these shapes / would like to see it!
Note: The square’s side lengths must be scaled up by sqrt(pi/2) but am interested to see it come about this way because of the geometric connection. Is there something special about the angle you have rotated through? Formula for the area of a square based on the angle and length of a transversal? Connection between multiplication as repeated addition (exhaustion to find pi) and rotation??
Apologies if this is already answered somewhere else in the site.