I have a quadrilateral with side lengths $10.40$, $12.33$, $11.75$, $11.50$. I am not given any other information, no angles or anything.
I do not need to find the area, since I know it is impossible with just the data given. Instead, I need to determine whether or not changing the order of the sides will affect the area of the quadrilateral.
More importantly - I need to know why it changes the area or why not.
It is possible to reorder the side lengths without changing the area -- namely, to interchange the lengths of two neighboring sides, mirror those two sides about the perpendicular bisector of the diagonal that connects their non-shared ends. This doesn't change the area.
If the original quadrilateral wasn't convex, you might end up with a butterfly shape where one of the triangles counts for negative area -- but there will always be some pair of sides you can swap without this happening, and from there you can always achieve your desired order of the sides by possibly mirroring the entire figure, and/or rotating it cyclically.
However, just from the specification "reorder the lengths", it is by no means a given that the reordering will happen in that particular way. (It probably won't, if you just tell someone to rearrange the sides and don't reveal that your goal is to keep the area unchanged).