Compare the areas of a square with perimeter $12$ and of a parallelogram with perimeter $16$

Question

Column A: The area of a square with perimeter $12$.

Column B: The area of a parallelogram with perimeter $16$.

Is Column A > B?, B < A, both equal, or is there not enough information?

The answer is that we cannot say anything (D), but I don't understand why.

Answer 1

We don't have enough information, because there are lots of parallelograms with perimeter 16.

First, note that the square with perimeter 12 has side length 3, and thus area 9.

Now consider first the square with side length 4; this is a parallelogram with perimeter 16, and area 16; its area is greater than the first square's. But let's say we chose the rectangle whose height was 1 and width was 7. This is also a parallelogram with perimeter 16, but its area is 7, which is decidedly less than 9.

So we've exhibited two different parallelograms with perimeter 16 - one with greater area than the first square, and one with smaller area. So we don't know whether the square has greater area or not unless we have more information about the parallelogram.

Answer 2

The answer is (D) because there isn't a method to find the area of a parallelogram of perimeter $16$, we need further information