area of the middle square

Let $x$ be the side length of the purple square, $y$ be the side length of the green square and $z$ be the side length of the blue square.

Since the red line is the diagonal of the purple square, its side length is

$x\sqrt{2}=15\sqrt{2}$

$x=15$

The side length of the green square is $y=\sqrt{16}=4$ .

Since the outside perimeter is $82$ , we have

$82=4+4+z-4+z+15-z+15+15+15+z+4$

$82=68+2z$

$z=7$

Therefore, the desired area is $7^2=\boxed{49}$ .