Let's solve this step-by-step:
1. Identify the given values:
- Central angle: [tex]\(\frac{2 \pi}{9}\)[/tex] radians
- Diameter of the circle: [tex]\(20.6\)[/tex] mm
- Value of [tex]\(\pi\)[/tex]: [tex]\(3.14\)[/tex]
2. Calculate the radius of the circle:
[tex]\[
\text{Radius} = \frac{\text{Diameter}}{2} = \frac{20.6}{2} = 10.3 \text{ mm}
\][/tex]
3. Calculate the area of the entire circle:
The formula to find the area of a circle is:
[tex]\[
\text{Area of the circle} = \pi \times \text{(radius)}^2
\][/tex]
Plugging in the values, we get:
[tex]\[
\text{Area of the circle} = 3.14 \times (10.3)^2 = 3.14 \times 106.09 = 333.1226 \text{ mm}^2
\][/tex]
4. Calculate the area of the sector:
The formula to find the area of a sector is:
[tex]\[
\text{Area of the sector} = \left(\frac{\text{Central angle in radians}}{2 \pi}\right) \times \text{Area of the circle}
\][/tex]
Substitute the given values into the formula:
[tex]\[
\text{Area of the sector} = \left(\frac{\frac{2 \pi}{9}}{2 \pi}\right) \times 333.1226 = \left(\frac{2 \pi}{9 \times 2 \pi}\right) \times 333.1226 = \left(\frac{1}{9}\right) \times 333.1226 = 37.01362222222223 \text{ mm}^2
\][/tex]
5. Round the answer to the nearest hundredth:
The area of the sector rounded to the nearest hundredth is:
[tex]\[
37.01 \text{ mm}^2
\][/tex]
Therefore, the area of the sector is [tex]\( \boxed{37.01} \)[/tex] mm[tex]\(^2\)[/tex].
