Let x be the length of a diagonal of a square of side $\text{'}a\text{'}$Then,

$\sqrt{2}a=x\Rightarrow a=\frac{x}{\sqrt{2}}$

Let A be the area of the square. Then, 

$A=a^2\Rightarrow A=\frac{x^2}{2}\Rightarrow \frac{dA}{dt}=x\frac{dx}{dt}$

It is given that $\frac{dx}{dt}=0.2$ and $\frac{dA}{dt}=6{\mathrm{cm}}^2/\sec$

$\therefore \frac{dA}{dt}=x\frac{dx}{dt}\Rightarrow 6=0.2x\Rightarrow x=30\mathrm{cm}\Rightarrow a=15\sqrt{2}\mathrm{cm}$