A wire of length 2 units is cut into two parts which are bent respectively to form a square of side = x…

Given that $4x+2\pi r=2$

i.e., $2x+\pi r=1$

$\therefore$ $r=\frac{1-2x}{\pi }$ .......(i)

Area $A=x^2+\pi r^2$

$=x^2+\frac{1}{\pi } {\left(2x-1\right)}^2$

For min vale of area $A$

$\frac{dA}{dx}=0$ gives $x=\frac{2}{\pi +4}$ .......(ii)

From (i) and (ii)

$r=\frac{1}{\pi +4}$ .......(iii)

$\therefore$ $x=2r$