To solve for the area of the sector given the arc length and the central angle, follow these steps:
1. Identify the given values:
- Arc length ([tex]\( L \)[/tex]) = [tex]\( \pi \)[/tex] cm
- Central angle ([tex]\( \theta \)[/tex]) = [tex]\( \frac{\pi}{6} \)[/tex] radians
2. Find the radius ([tex]\( r \)[/tex]) of the circle using the arc length formula:
The formula for the arc length of a sector is:
[tex]\[
L = r \theta
\][/tex]
Substituting the given values:
[tex]\[
\pi = r \cdot \frac{\pi}{6}
\][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[
r = \frac{\pi}{\frac{\pi}{6}} = \frac{\pi \times 6}{\pi} = 6 \text{ cm}
\][/tex]
3. Calculate the area of the sector:
The formula for the area ([tex]\( A \)[/tex]) of a sector is:
[tex]\[
A = \frac{1}{2} r^2 \theta
\][/tex]
Substituting the radius and central angle:
[tex]\[
A = \frac{1}{2} \times 6^2 \times \frac{\pi}{6}
\][/tex]
Simplify the expression:
[tex]\[
A = \frac{1}{2} \times 36 \times \frac{\pi}{6} = \frac{36 \pi}{12} = 3 \pi \text{ cm}^2
\][/tex]
Therefore, the area of the sector is [tex]\( 3 \pi \)[/tex] cm[tex]\(^2\)[/tex].
Thus, the correct answer is:
D. [tex]\( 3 \pi \)[/tex] cm[tex]\(^2\)[/tex]
