## Task 2 - 090822

On a straight line, the points $A, E, C, F, O, G, D$, $H, B$ are arranged in this order such that:

\[
\begin{aligned}
& \overline{A E}=\overline{E F}=\overline{F O}=\overline{O B}=6 \mathrm{~cm} \\
& \overline{E C}=\overline{C F}=\overline{G D}=\overline{H B}=1 \mathrm{~cm} \\
& \overline{D H}=0.5 \mathrm{~cm}
\end{aligned}
\]

Semicircles are drawn over the segments $A B, E H$ and $F G$ in one half-plane, and over the segments $A O, O B, E F$ and $G H$ in the other half-plane with respect to the line through $A$ and $B$.

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0631.jpg?height=435&width=636&top_left_y=1333&top_left_x=1153)

Calculate the area of the colored region!
The sought area $A$ is equal to the sum of the areas of the semicircles over $AB, FG, AO$, and $OB$ minus the sum of the areas of the semicircles over $EH, EF$, and $GH$.

Given $AB=6 \text{ cm}, FG=2 \text{ cm}, AO=OB=3 \text{ cm}, EH=4 \text{ cm}, EF=GH=1 \text{ cm}$, it follows that:

$$
A=\frac{\pi}{8}(36+4+9+9-16-1-1) \text{cm}^{2}=5 \pi \text{cm}^{2}
$$

The area of the shaded region is $5 \pi \text{cm}^{2}$, which is approximately $15.71 \text{ cm}^{2}$.