The area of a sector of a circle having radius $14\text{cm}$ is $231\text{cm}{2}^{}$. Find the degree measure of the corresponding central angle. (Use $\pi =\frac{22}{7}$)

The correct answer is 135°.

We are given:

Radius, $r=14$ cm
Area of sector, $A=231$ cm²
$\pi =\frac{22}{7}$

We need to find the central angle $\theta$ (in degrees).

Step 1: Recall the formula for the area of a sector.

$A=\frac{\theta }{360}\times \pi \times r^2$

where $\theta$ is the central angle in degrees.

Step 2: Substitute the known values into the formula.

$231=\frac{\theta }{360}\times \frac{22}{7}\times {14}^2$

Step 3: Simplify ${14}^2=196$.

$231=\frac{\theta }{360}\times \frac{22}{7}\times 196$

Step 4: Simplify $\frac{22}{7}\times 196$.

Since $\frac{196}{7}=28$, we get:

$\frac{22}{7}\times 196=22\times 28=616$

So the equation becomes:

$231=\frac{\theta }{360}\times 616$

Step 5: Solve for $\theta$.

$\theta =\frac{231\times 360}{616}$

First, simplify $\frac{360}{616}$ by dividing both by 8:

$\frac{360}{616}=\frac{45}{77}$

So:

$\theta =231\times \frac{45}{77}$

Simplify $\frac{231}{77}=3$:

$\theta =3\times 45=135$

The degree measure of the central angle is 135°.