A wire is bent in the form of a square of area $121\text{ cm}^2$. If the same wire is bent in the form of a circle, what is the area of the circle?
Method 1 (Traditional Logical Step)
- Find the side of the square ($a$): $Area = a^2 = 121 \Rightarrow a = 11\text{ cm}$.
- Find the length of the wire (Perimeter of Square): $P = 4a = 4 \times 11 = 44\text{ cm}$.
- Circumference of Circle = Length of Wire: $2\pi r = 44 \Rightarrow 2 \times \frac{22}{7} \times r = 44 \Rightarrow r = 7\text{ cm}$.
- Find Area of Circle: $Area = \pi r^2 = \frac{22}{7} \times 7 \times 7 = \mathbf{154\text{ cm}^2}$.
Method 2 (Booster Shortcut) ⚡
When a wire is bent from a Square to a Circle (or vice versa), the perimeter remains constant. You can use the direct Constant Perimeter Ratio:
$$\frac{\text{Area of Circle}}{\text{Area of Square}} = \frac{14}{11}$$
(This ratio is derived from $\frac{\pi r^2}{(\frac{2\pi r}{4})^2}$)
Calculation:
$$\text{Area of Circle} = 121 \times \left(\frac{14}{11}\right) = 11 \times 14 = \mathbf{154\text{ cm}^2}$$
Time taken: 5 seconds!
Correct Answer: A) 154 cm²
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