Question

Shown in the given figure is a circle with centre \(O\). The area of the minor sector is \(7 \text{ cm}^{2}\). Area of circle is :

• A. \(84 \pi \text{ cm}^{2}\)
• B. \(\frac{84}{11} \text{ cm}^{2}\)
• C. \(84 \text{ cm}^{2}\)
• D. \(\frac{\sqrt{84}}{\sqrt{\pi}} \text{ cm}^{2}\)

Hint:

If the sector angle is \(30^{\circ}\), there are \(360/30 = 12\) such sectors in a circle. Just multiply the sector area by 12 to get the full circle area.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The area of a sector is a fraction of the total area of the circle, proportional to the central angle \(\theta\).
Step 2: Key Formula or Approach:
\[ \text{Area of Sector} = \frac{\theta}{360^{\circ}} \times \text{Area of Circle} \]
Step 3: Detailed Explanation:
From the figure, the central angle \(\theta = 30^{\circ}\).
Given Area of Sector = 7 \text{ cm}\^{2}.
\[ 7 = \frac{30^{\circ}}{360^{\circ}} \times \text{Area of Circle} \]
\[ 7 = \frac{1}{12} \times \text{Area of Circle} \]
Multiply both sides by 12:
\[ \text{Area of Circle} = 7 \times 12 = 84 \text{ cm}^{2} \]
Step 4: Final Answer:
The total area of the circle is 84 \text{ cm}\^{2}.