If a circle and a square have the same perimeter, then
their areas are equal
the area of the circle is greater than the area of the square
the area of the square is greater than the area of circle
the area of the circle is two times the area of the square
Answer is Right!
Answer is Wrong!
This question was previously asked in
UPSC CAPF – 2019
For a square with side length $s$, the perimeter is $4s=P$, so $s = P/4$. The area of the square is $A_{\text{square}} = s^2 = (P/4)^2 = P^2/16$.
For a circle with radius $r$, the perimeter is $2\pi r=P$, so $r = P/(2\pi)$. The area of the circle is $A_{\text{circle}} = \pi r^2 = \pi (P/(2\pi))^2 = \pi (P^2/(4\pi^2)) = P^2/(4\pi)$.
Since $\pi \approx 3.14159$, $4\pi \approx 12.566$.
Comparing $1/16$ and $1/12.566$. Since $16 > 12.566$, it follows that $1/16 < 1/12.566$. Therefore, $A_{\text{square}} < A_{\text{circle}}$. The area of the circle is greater than the area of the square. This is a general geometric principle: among all planar shapes with the same perimeter, the circle has the largest area.