Question

If the area of a sector is one-twelfth that of a complete circle, then the angle of the sector is :

• A. 36°
• B. 30°
• C. 60°
• D. 45°

Answer 1

The Correct Option is B

Step 1: Understanding the problem:
We are asked to find the angle of a sector whose area is one-twelfth of the area of a complete circle.
Let the radius of the circle be \(r\) and the central angle of the sector be \(\theta\) (in radians).

Step 2: Formula for the area of a sector:
The area \(A_{\text{sector}}\) of a sector of a circle with radius \(r\) and central angle \(\theta\) is given by the formula: \[ A_{\text{sector}} = \frac{1}{2} r^2 \theta \] The area \(A_{\text{circle}}\) of a complete circle with radius \(r\) is: \[ A_{\text{circle}} = \pi r^2 \]

Step 3: Relating the areas:
We are told that the area of the sector is one-twelfth of the area of the complete circle: \[ A_{\text{sector}} = \frac{1}{12} A_{\text{circle}} \] Substitute the formulas for \(A_{\text{sector}}\) and \(A_{\text{circle}}\): \[ \frac{1}{2} r^2 \theta = \frac{1}{12} \pi r^2 \] Cancel out \(r^2\) from both sides: \[ \frac{1}{2} \theta = \frac{1}{12} \pi \] Multiply both sides by 2 to isolate \(\theta\): \[ \theta = \frac{1}{6} \pi \]

Step 4: Conclusion:
Thus, the angle of the sector is \(\frac{\pi}{6}\) radians or 30°.