To solve the problem, we need to determine the ratio of the area of the sector [tex]\(AOB\)[/tex] to the area of the entire circle when given that the ratio of the arc length [tex]\(\hat{AB}\)[/tex] to the radius of the circle is [tex]\(\frac{\pi}{10}\)[/tex].
1. Understanding the Given Ratio:
The given ratio is:
[tex]\[
\frac{\text{arc length of } \hat{AB}}{\text{radius}} = \frac{\pi}{10}
\][/tex]
According to the geometry of the circle, the arc length of [tex]\( \hat{AB} \)[/tex] is also given by:
[tex]\[
\text{Arc length of } \hat{AB} = r \theta
\][/tex]
where [tex]\(r\)[/tex] is the radius and [tex]\(\theta\)[/tex] is the central angle in radians subtended by the arc. Thus:
[tex]\[
\frac{r \theta}{r} = \frac{\pi}{10} \implies \theta = \frac{\pi}{10}
\][/tex]
2. Determining the Ratio of the Areas:
The area of a sector of a circle is given by:
[tex]\[
\text{Area of sector } = \frac{1}{2} r^2 \theta
\][/tex]
and the area of the entire circle is:
[tex]\[
\text{Area of circle} = \pi r^2
\][/tex]
The ratio of the area of sector [tex]\(AOB\)[/tex] to the area of the circle is:
[tex]\[
\frac{\text{Area of sector}}{\text{Area of circle}} = \frac{\frac{1}{2} r^2 \theta}{\pi r^2 }
\][/tex]
3. Simplifying the Expression:
Substituting [tex]\(\theta = \frac{\pi}{10}\)[/tex] into the ratio:
[tex]\[
\frac{\frac{1}{2} r^2 \frac{\pi}{10}}{\pi r^2} = \frac{1}{2} \cdot \frac{\pi}{10} \cdot \frac{1}{\pi} = \frac{1}{2} \cdot \frac{1}{10} = \frac{1}{20}
\][/tex]
Thus, the ratio of the area of sector [tex]\(AOB\)[/tex] to the area of the circle is:
[tex]\[
\boxed{\frac{1}{20}}
\][/tex]
So, the correct answer is C. [tex]\(\frac{1}{20}\)[/tex].
