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A sector of a circle has an arc length of [tex]\pi \, \text{cm}[/tex] and a central angle of [tex]\frac{\pi}{6}[/tex] radians. What is the area of the sector?

A. [tex]\pi \, \text{cm}^2[/tex]

B. [tex]5\pi \, \text{cm}^2[/tex]

C. [tex]2\pi \, \text{cm}^2[/tex]

D. [tex]3\pi \, \text{cm}^2[/tex]



Answer :

GinnyAnswer
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]

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