Question

A circle and a square have the same perimeter. Then

• A. Their areas are equal
• B. The area of the circle is greater
• C. The area of the square is greater
• D. None of the above

Answer 1

Answer: (B)

The area of the circle is greater

Let the length of a side of square be $a$ and the radius of the circle is $r$.

Since square and circle have the same perimeter.
Therefore, $2\pi r=4a$
$\Rightarrow r=\frac{2a}{\pi }$
Thus $\frac{\text{Area of the circle}}{\text{Area of the square}}=\frac{\pi r^2}{a^2}=\frac{\pi \cdot \frac{4a^2}{{\pi }^2}}{a^2}=\frac{4}{\pi }>1$
Hence, the area of the circle is greater than area of the square.