In that case, the radius of largest circle would be the hypotenuse & square's half sides would be the base & perimeter.
So, H² = 0.5²+0.5²
H² = 0.25 + 0.25
H² = 0.50
H = √0.50
H = 0.707 of unit value.
Radius & your final answer would be 0.707 of unit value.
Hope this helps!