The radius of the small circle = 20-=20-20sin45 /sin67.5. The source of this formula is that the intersection of a small circle and a large arc is the midpoint of the large arc, and the length of the connecting line between the intersection and the square is equal to the edge of the square MINUS the radius of the small circle.

Circumference = 2π (20-20 sin45/sin67.5)

Assuming the scheme is feasible, 2 π (20-20 sin45/sin67.5) =10 π, 3/4 = sin45/sin67.5.

All you have to do is prove that the above formulas are not equal.

Scheme 4: it is necessary to set the arch height of the cone =x and the radius of the cone = √ [10 2+(20-x) 2].

Arc length = 2atan [10/(20-x)] * √ [10 2+(20-x) 2],

The radius of small circle r:10-r+20-r-x = √ [102+(20-x) 2] = 30-2r-x,

r^2+r(x-30)+ 100-25x=0,r=[30-x+-√(x^2+40x+500)]/2

Circumference = π [30-x+-√ (x 2+40x+500)], and then compare arc length = 2atan [10/(20-x)] * √ [10 2+(20-x) 2].

How annoying! Or is there something wrong with my method?