Question

11 a central angle of 120° subtends an arc in a circle with radius 6 units. find the arc - sector area.
a. 12π square units
b. 18π square units
c. ⅓ square units
d. ⅔ square units

Explanation:

Step1: Convert angle to radians

The formula to convert degrees to radians is $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. Given $\theta_{deg} = 120^{\circ}$, then $\theta_{rad}=120\times\frac{\pi}{180}=\frac{2\pi}{3}$.

Step2: Use sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius of the circle and $\theta$ is the central - angle in radians. Here, $r = 6$ and $\theta=\frac{2\pi}{3}$. Substitute these values into the formula: $A=\frac{1}{2}\times6^{2}\times\frac{2\pi}{3}$.

Step3: Calculate the area

First, $6^{2}=36$. Then $\frac{1}{2}\times36\times\frac{2\pi}{3}=12\pi$.

Answer:

A. $12\pi$ square units