geometry

From the center of the  small circle  draw a perpendicular to  the bottom  edge of the  square

  .....the distance = 4-r   = A

From  the  center of the semi-circle draw a segment to the point where the first segment intersects the base

.....this distance is 2-r  = B

Connect the center of the  small circle to the center of the semi-circle....the distance is 2 + r = C

We have a right triangle  such that

C^2 = A^2 + B^2

(2 + r)^2  = (4- r)^2  + (2-r)^2    simplify

r^2 + 4r + 4 =  r^2 -8r + 16  + r^2 - 4r + 4

r^2 + 4r + 4  =  2r^2 - 12r + 20

r^2  - 16r + 16  =  0

r^2 -16r   =  -16

r^2 -16r  + 64   = -16 + 64

(r - 8)^2  =  48

r - 8  =   sqrt (48)           or      r - 8 = -sqrt (48)

The first value for r is too large

The second value produces   r =  8 -sqrt (48)  ≈  1.07