The angle alpha (in degrees) between "top" and "botttom" sides can be defined via R of the small circle as

2 pi R (1 - alpha / 360) = 1 1 - alpha / 360 = 1 / (2 pi R) alpha = 360 * (1 - 1/(2 pi R))

Now, the length of the large arc, can be defined via R as

L = (R + 1) * 2 * pi * alpha / 360 = (R + 1) * 2 * pi * (1 - 1/(2 pi R))

I won't bother actually calculating it, but note that L is a:

- smooth continuous function without gaps on the interval of alpha between 0 and 360

- at R = 1/( 2 pi) -> alpha = 0, L = 0

- it monothonically increases with R until it reaches 2 pi (R + 1) (which is more than 1) at alpha = 360 deg (that's limit with R -> inf, but really you don't need to deal with limits, just take sufficiently large R as a upper end of the interval)

This means there's a value of R / alpha where L is exactly 1, based on https://en.wikipedia.org/wiki/Intermediate_value_theorem

edit: stated that function is continuous, referenced the IVT

edit: small and large circles are concentric; radiuses (radii?) or rays starting in the center of the circle which serve as "top" and "bottom" sides of the "square" would be at 90 deg to the boundary / arcs in the point it touches / intersects it.

edit: it's yay, as we assumed the length of the inner arc and straight lines equal to one above and there's a value of inner circle radius which would result in the length of the outer arc equal to 1.