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: Think in terms of equal slices instead of formulas.
A sector with a $30^\circ$ central angle is one of $\frac{360^\circ}{30^\circ}=12$ identical slices that together make up the whole circle.
Step 2: Use the fact that all 12 slices are congruent.
Equal central angles mean equal sector areas, so the full circle's area is exactly 12 times the area of one such slice.
Step 3: Compute.
Area of circle $=12\times7=84$ cm$^2$.
\[ \boxed{84 \text{ cm}^2} \]