Square in Square

let x = length of side of the shaded square

$x=\sqrt{1^2+2^2}$

$x=\sqrt{5}$

$area of shaded square=(\sqrt{5})^2$ $=5$

$area of large square=3^2=9$

$area of shaded square=\frac{5}{9}area of large square$

You can see that the red are is the area (1) of the central square plus half the area of the 8 remaining squares (4).

And since there are 9 squares in total the ratio is $\dfrac{1+4}{9} = \dfrac{5}{9}$