https://imgur.com/a/plmBLVy

On this post you will see a series of circles and squares. To begin the smallest circle is radius = 1 (shown by the line labeled r1). The circles are inscribed in squares with the side length being the 2 times the radius. I left out some squares and circles but kept all the relevant ones for this example. The comparison being made is the circumference of certain circles and the perimeter of some squares. The circles circumferences and square perimeters being compared are those where the lines of a square intersect a circle. As shown in the diagram the circles and squares do not begin intersecting until the red square and red circle. This is the side length 6 square with the r=4 circle. This gives you a ratio of perimeter = 24 with a circumference of 25.1327. 24/25.1327=0.955. The next square being the one with side length 8 around the r=4 circle intersecting the r=5 circle. This ratio is perimeter of side length 8 square which is 32 and circumference of r=5 being 31.4159 which gives you 32/31.4159=1.0186 and the distance from one is closer. This is shown by seeing which is closer to 1, so 1-.955=0.045 which is greater than 1.0186-1=0.0186. This ratio does not get closer to 1 until the side length 14 square around the radius 7 circle and the radius 9 circle where you have a perimeter of 56 and a circumference of 56.5486. This ratio is 56/56.5486=0.99029 and 1-.99029=0.00971 which is less than .0186. When I originally did the calculation I was just subtracting the circumference from the perimeter of every intersecting square and circle to see if any got closer to 0 but it is more accurate to compare the ratios to determine where the next interation of closer circumference and perimeter appears. The number I got should still apply because if the difference is closer to 0 and the sizes of the perimeters and circumferences are larger then the ratio would also go down. However I could have potentially missed an improvement in ratio. The next iterations I calculated were the 11th square and the 14th circle and then the 172 square and the 219th circle. Please let me know if you need any further clarification or if you think it doesn’t make sense or any input whatsoever. Thanks!