Question : If the circumference of a circle is equal to the perimeter of a square, and the radius of the given circle is positive, then which of the following options is correct?
Option 1: Area of the circle > Area of the square
Option 2: Area of the circle $\geq$ Area of the square
Option 3: Area of the circle < Area of the square
Option 4: Area of the circle = Area of the square
Correct Answer: Area of the circle > Area of the square
Solution :
Let the radius of the circle be '$r$' and side of the square be 'a'
Since the perimeter of a square = circumference of a circle
⇒ 4a = 2$\pi r$
⇒ a = $\frac{\pi r}{2}$
⇒ Area of Square = $(\frac{\pi r}{2})^{2}$
⇒ $\frac{\text{Area of circle}}{\text{Area of square}} = \frac{\pi r^{2}}{(\frac{\pi r}{2})^{2}}=\frac{4}{\pi}$ > 1
So, the area of the circle > the area of the square.
Hence, the correct answer is Area of the circle > Area of the square.
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