If a circle and a square have the same perimeter, then
[amp_mcq option1=”their areas are equal” option2=”the area of the circle is greater than the area of the square” option3=”the area of the square is greater than the area of circle” option4=”the area of the circle is two times the area of the square” correct=”option2″]
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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.