Find ratio of areas in circles and rectangles

[Okkk]

Let the diameter of the circle be R and the radius of the quarter circles be r. Then This relation can be inferred from a right triangle height (r) division ratio of the hypotenuse (2R) considering triangle similarities. Now the required answer is

[TATARLaine]

Well I found an other result and I cannot find where I mistook... here is my calculation
Lets call a the length AD, b the length DC c the length AC r the length FD (radius of small green circle).
Finally x the angle CAD. The goal is to express all areas in terms of b and b only.
tan(x)=FD/AF=3FD/2AC= 3r/2c it is also tan(x)=FC/FD=AC/3FD = c/3r so tan(x)^2 = 1/2.
the relation between a and b (AD and CD) is therefore b/a=tan(x) => b=a/sqrt(2) => b^2 = a^2/2 => a^2= 2*b^2
so the large circle area is (a/2)^2*PI= b^2/2*PI
for the small circle the radius r is given by DC*cos(x) and its area is PI*(CD)^2*cos(x)^2
tan(x)^2 = sin(x)^2/cos(x)^2 = => cos(x)^2 = 2/3
so total area of a green circle is given by PI*b^2/3
we have two quarters of green circle, which gives a green area of PI*b^2/6
so the ratio is (PI*b^2/6)/(b^2/2*PI)=1/3
Where is the mistake, if someone can spot it out, I would be grateful