Semicircles in a Circle

For simplicity just reverse the lower half of the given figure you will see $3$ complete circles.

let $c_1$ be smallest circle with radius $r_1$

$c_2$ be tha middle circle with radius $r_2$

and $c_3$ be the largest circle with radius $r_3$

, since it is given that $AB=BC=CD$ so we can take that $r_3=3r_1~and r_2=2r_1$ . for $r_1=1$

therefore, $area ~of~ shaded~ region~=area ~of ~c_3+ ~area~of~c_1-area~of~c_2=$ $=\pi \times (3^2 + 1^2 - 2^2) = 6\pi$

and $total ~area= 9\pi$

$therefore~ proportion ~of ~shaded~ region = \dfrac{6\pi}{9\pi} =\dfrac{2}{3}$