In this question, we are comparing the perimeter and area of a circle and a square. Let`s analyze each option to understand the correct answer.

Option 1: If a circle and a square have the same perimeter, it does not imply that their areas are equal. The formula to calculate the area of a circle is π * r^2, and for a square, it is side^2. Since the perimeter is equal, the lengths of their sides will be different, resulting in different areas.

Option 2: This is the correct answer. If the perimeter of the circle and the square is the same, the circle will have a greater area. This is because a circle utilizes its perimeter efficiently, while a square has corners that reduce its area.

Option 3: If the perimeter of the circle and the square is the same, it does not imply that the area of the square is greater. The circle can still have a greater area because it uses its perimeter efficiently.

Option 4: If the perimeter of the circle and the square is the same, the area of the circle cannot be exactly two times the area of the square. This relationship depends on the dimensions of the square and the circle, and it cannot be generalized.

So, option 2 is the