Calculating Sector Angle Using Area and Radius

We are given the radius of a sector, $r = 14 \text{ cm}$, and its area, $A = 154 \text{ cm}^2$. We need to find the angle of the sector, $\theta$.

Sector Area Formula

The formula for the area of a sector is:

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

Where:

• $A$ is the area of the sector.
• $\theta$ is the angle of the sector in degrees.
• $r$ is the radius of the sector.
• $\pi$ is the mathematical constant Pi (approximately $\frac{22}{7}$).

Applying the Formula

Substitute the known values into the formula:

$ 154 = \frac{\theta}{360^\circ} \times \frac{22}{7} \times (14)^2 $

Solving for the Angle ($\theta$)

1. Calculate $r^2$: $(14)^2 = 196$.
2. Substitute this back into the equation:

   $ 154 = \frac{\theta}{360^\circ} \times \frac{22}{7} \times 196 $

3. Simplify the expression $\frac{22}{7} \times 196$:

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

4. The equation becomes:

   $ 154 = \frac{\theta}{360^\circ} \times 616 $

5. Rearrange to solve for $\theta$:

   $ \theta = \frac{154 \times 360^\circ}{616} $

6. Simplify the fraction. Notice that $616 = 4 \times 154$:

   $ \theta = \frac{154 \times 360^\circ}{4 \times 154} $

7. Cancel out 154:

   $ \theta = \frac{360^\circ}{4} $

8. Calculate the final angle:

   $ \theta = 90^\circ $

Conclusion

The angle of the sector is $90^\circ$. This corresponds to Option A.