How can two rectangles have the same area and different perimiters?

Answer 1

It could happen. For Example: If one of the sides is 4 inches and the other is 3 inches--

Area: 12

Perimeter: 18

But if one side is 2 inches and the other is 6 inches:

Area: 12

Perimeter: 16

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I think the questioner knows it happens, and is wondering why it happens.

You may have heard this, but it will make sense intuitively even if you haven't. A soap bubble is a sphere because it is the natural shape that has the smallest surface area that the gas within can take. What force would there normally be keeping a single free-floating bubble in any shape other than a sphere? In your mind, stretch a bubble out to be any shape other than a sphere, and you will sense that the new shape can't be sustained by nature; natural forces will bring the bubble back to a sphere. Under ordinary circumstances you have never seen a bubble that is not a sphere, or that is not tending toward a sphere-like shape given other limiting factors, like the bubble resting on the surface of water, or bubbles grouped together.

It won't be surprising then that a circle of a given area will have a circumference a little smaller than the perimiter of a square with the same area. A circle with radius=1 will have an area of pi square units. A square that has one side equal to the square root of pi will have an area of pi square units. The circumference of a circle is equal to pi times the diameter (2 X the radius). So our circle has a circumference of 6.28318. The square of equal area will have a perimiter of 4(square root of pi), or 7.08981. You will get similar results comparing the surface areas of a sphere and a cube of equal volume.

In the plane geometry of rectangles exclusively, a square with 32 units on one side will have an area of 1024 square units. If a rectangle has 8 units on one side the other side has to be 128 units to come to an area of 1024, for a perimiter of 272 units. If a rectangle is 4 units on one side, the other side must be 256 units to come to an area of 1024, and the perimiter is 520. If one side is .5 units long, the other side must be 2048 units, and the perimiter is 4097 units!

Think of it this way. Visualize a square. In the square, every unit of width distributes the same 'area' across every unit of height, and every unit of height distributes the same 'area' across every unit of width; there is an efficient distribution of area across dimensions (if you want your final shape to remain a square), not unlike a bubble that pulls itself by natural forces into the smallest ratio of surface area to volume.

Another way of thinking about this: Length of perimiter is, obviously, a linear measure-- you are adding the lengths of lines. Lines have one dimension in Euclidean geometry. Take a line parallel to the y axis and that is arbitrarily long (as long as you want to imagine it). So far there is no area whatsover, right? By extending a line from the base and parallel to the x axis, you can define a rectangle with an area arbitrarily close to zero, or approaching infinity. The area of a shape is not necessarily proportional to the lengths of any of the sides of the shape.