The correct answer is (a).
The perimeter of a circle is $2\pi r$, where $r$ is the radius of the circle. The perimeter of a square is $4s$, where $s$ is the side length of the square.
If a circle and a square have the same perimeter, then $2\pi r = 4s$. Solving for $r$, we get $r = \frac{2s}{\pi}$.
The area of a circle is $\pi r^2$. The area of a square is $s^2$.
Substituting $r = \frac{2s}{\pi}$ into the formula for the area of a circle, we get $\pi \left( \frac{2s}{\pi} \right)^2 = \frac{4s^2}{\pi}$.
Substituting $r = \frac{2s}{\pi}$ into the formula for the area of a square, we get $s^2$.
Therefore, the area of the circle is equal to the area of the square.
Option (b) is incorrect because the area of a circle is always less than or equal to the area of a square with the same perimeter. This is because the area of a circle is proportional to the square of its radius, while the area of a square is proportional to the square of its side length. Since the radius of a circle is always less than or equal to the side length of a square with the same perimeter, the area of a circle is always less than or equal to the area of a square with the same perimeter.
Option (c) is incorrect because the area of a square is always greater than or equal to the area of a circle with the same perimeter. This is because the area of a square is proportional to the square of its side length, while the area of a circle is proportional to the square of its radius. Since the side length of a square is always greater than or equal to the radius of a circle with the same perimeter, the area of a square is always greater than or equal to the area of a circle with the same perimeter.
Option (d) is incorrect because the area of a circle is never two times the area of a square with the same perimeter. This is because the area of a circle is proportional to the square of its radius, while the area of a square is proportional to the square of its side length. Since the radius of a circle is always less than or equal to the side length of a square with the same perimeter, the area of a circle is always less than or equal to the area of a square with the same perimeter.