A Problem From My Exam In the image, a semicircle with diameter $AD$ has smaller semicircles with diameters $AB$, $BC$, $CD$, all next to each other inside it. The area between them is shaded.
The perimeter of the painted area is $24\pi$, what is the area?


Ratio of diameters
$$
\begin{align}
AB : BC : CD = 1 : 2 : 3 =
r_{1} : r_{2} : r_{3}
\end{align}
$$
Radius of perimeter segment: $\pi r = 24 \pi$.
$$ 
 r = 24
$$
Add up the sgements:
$$
 \begin{align}
  r &= r_{1} + r_{2} + r_{3} \\
    &= r_{1} + 2 r_{1} + 3 r_{1} = 6 r_{1} \\
  24 &= 6 r_{1} \\
   4 &= r_{1}
 \end{align}
$$

Shaded area
$$
 \begin{align}
%
A &= \frac{\pi}{2} \left( r^{2} - 
\left( 
  r^{2}_{1} + r^{2}_{2} + r^{2}_{3}
\right)
\right) \\
%
&=\frac{\pi}{2} \left( 24^{2} - 
\left( 
  4^{2} + 8^{2} + 12^{2}
\right)
\right) \\
&= 352 \frac{\pi}{2} = \boxed{176 \pi}
%
 \end{align}
$$
