A circular region has a radius measuring 12 cm. Determine the area of the sector formed by a central angle of $\frac{\pi }{5}$ radians.

The correct answer is 45.24 cm².

Given Data:

Radius of the circular region, $r=12\text{cm}$

Central angle of the sector, $\theta =\frac{\pi }{5}\text{radians}$

Formula:

The area of a sector of a circle when the central angle $\theta$ is measured in radians is given by the formula:

$A=\frac{1}{2}{r}^2\theta$

Step-by-Step Calculation:

Step 1: Substitute the given values into the formula

$A=\frac{1}{2}\times {12}^2\times \frac{\pi }{5}$

Step 2: Square the radius

${12}^2=144$

Step 3: Simplify the numerical expression

$A=\frac{1}{2}\times 144\times \frac{\pi }{5}$

$A=72\times \frac{\pi }{5}$

$A=14.4\pi$

Step 4: Calculate the decimal value using $\pi$

Substituting $\pi \approx 3.14159$:

$A\approx 14.4\times 3.14159$

$A\approx 45.2389\text{cm}{2}^{}$

Step 5: Round to two decimal places

Rounding $45.2389$ to two decimal places gives $45.24$.

Therefore, the area of the sector is 45.24 cm².