Let r = radius of inner circle.

The radius R of the outer circle is a diagonal of any of the 4 squares that make up the one big square.

And this, by Pythagorean theorem is

R^2 = r^2 +r^2

R^2 = 2r^2 -----------------***

Big circle,

Area = pi(R)^2 = 256pi

pi(2r^2) = 256pi

2r^2 = 256

r^2 = 256/2

r^2 = 128

Small circle,

Area = pi(r^2) = pi(128) = 128pi sq.mm -------------answer.