Answer:
1.6
Explanation:
You want the ratio of the side of a square to the radius of a circle that is tangent to one side and passes through the end points of the opposite side of the square.
Circle
The radius of the circumcircle of the isosceles triangle inscribed in a square is 5/8 of the length of the side of the square. This is seen in the attached figure, which is drawn to scale.
Circumcenter
For a unit square located so the midpoint of the top side is the origin, the bottom right corner is located at (1/2, -1). The perpendicular bisector of the line segment between the origin and that point will have the equation ...
y+1/2 = 1/2(x -1/4)
The y-intercept of this line is the opposite of the circle's radius:
y = 1/2x -5/8
The circle radius is 5/8.
Ratio
The ratio of side length to radius is ...
s/r = 1/(5/8) = 8/5 = 1.6 . . . . exactly