Let length of the rectangle be $2a$.

And let the breadth be $b$.

So, area $A=2ab$.

Now, we have, $a^2+b^2=1$.

$\Rightarrow A=2b\sqrt{1-b^2}$

Differentiating w.r.t. $b$ we get,

$\frac{dA}{db}=2\sqrt{1-b^2}+\frac{2b\left(-2b\right)}{2\sqrt{1-b^2}}$

Putting, $\frac{dA}{db}=0\Rightarrow b^2=\frac{1}{2}\Rightarrow b=\frac{1}{\sqrt{2}}$.