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:

A sector of $30^\circ$ is exactly $\frac{30^\circ}{360^\circ} = \frac{1}{12}\text{th}$ of the entire circle.
Therefore, the total area of the circle must be exactly 12 times the area of this sector.
\[ 7 \times 12 = 84 \text{ cm}^2 \]
No need to calculate the radius $r$ or use the value of $\pi$!

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given a circle with centre O. The area of the minor sector SOT subtending an angle of $30^\circ$ is $7 \text{ cm}^2$.
We need to determine the total area of this circle.

Step 2: Key Formula or Approach:
The area of a sector of a circle with radius $r$ and sector angle $\theta$ is given by:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \pi r^2 \]
Since the total area of the circle is $A = \pi r^2$, we can rewrite this relationship as:
\[ \text{Area of Sector} = \frac{\theta}{360^\circ} \times \text{Area of Circle} \]

Step 3: Detailed Explanation:

• Identify the given parameters from the problem description and the figure:
- Sector angle, $\theta = 30^\circ$
- Area of the minor sector $= 7 \text{ cm}^2$

• Substitute these values into the sector area formula:
\[ 7 = \frac{30^\circ}{360^\circ} \times \text{Area of Circle} \]

• Simplify the fraction representing the ratio of the sector angle to the full rotation:
\[ \frac{30^\circ}{360^\circ} = \frac{3}{36} = \frac{1}{12} \]

• Rewrite the equation with the simplified fraction:
\[ 7 = \frac{1}{12} \times \text{Area of Circle} \]

• Solve for the total Area of the Circle by multiplying both sides by 12:
\[ \text{Area of Circle} = 7 \times 12 \]
\[ \text{Area of Circle} = 84 \text{ cm}^2 \]


Step 4: Final Answer:
The total area of the circle is $84 \text{ cm}^2$.
Hence, option (C) is correct.