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Infinitely many. Suppose the area of the rectangle is 100. We could create rectangles of different areas:

100x1

50x2

25x4

20x5

10x10

However, the side lengths need not be integers, which is why we can create infinitely many rectangles.

Generally, if A is the area of the rectangle, and L, L/A are its dimensions, then the amount 2(L + (L/A)) can range from a given amount (min. occurs at L = sqrt(A), perimeter = 4sqrt(A)) to infinity.

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Q: How many rectangles have the same area but different perimeters?
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