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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.

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

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thare is only 1 differint rectangles

They can, but not necessarily. Let's say we have a rectangle of 5x6 and another of 1x30. The areas of both are 30 square feet, but their perimeters are different. Perimeter of 5x6: 5+5+6+6=22 feet Perimeter of 1x30: 1+1+30+30=62 feet There are many more examples of this.

18x1 9x2 6x3

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The area doesn't tell you the perimeter. There are an infinite number of different shapes and different dimensions that all have the same area but different perimeters. Here are a few 1/4-acre rectangles, and their perimeters: 90-ft x 121-ft . . 422-ft 30' x 363' . . . . . 786' 15' x 726' . . . . 1,482' 10' x 1,089' . . . 2,198' 6' x 1,815' . . . . 3,642' 2' x 5,445'. . . . 10,894'

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Infinite amounts.

technically the number is infinite

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3 or 6, depending on whether rectangles rotated through 90 degrees are counted as different. The rectangles are 1x12, 2x6 3x4 and their rotated versions: 4x3, 6x2 and 12x1.

No. Many investigators have searched for such an example, but none have found it yet. According to all published research so far, two rectangles with the same area always have the same area. But the search goes on, in many great universities.

You can't tell the shape or the distance around from the area. There are an infinite number of shapes and perimeters that all have areas of 100 acres. The shortest perimeter possible is a circle, with diameter (distance across) of 2,355 feet. The distance around it is 7,398 feet (about 1.4 miles). The shortest possible perimeter with straight sides is a square, 2,087 feet on every side. The distance around it is 8,348 feet (about 1.58 miles). If you stay with rectangles, there are an infinite number of different rectangles with different dimensions and different perimeters, that all have areas of 100 acres. The distance around every one of them is larger than 1.58 miles, and it can be anything up to infinity.

thare is only 1 differint rectangles

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