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JCX's avatar

I really am no mathematician but let me give it a go.

For any given perimeter, a circle will always be the most efficient way to enclose the biggest possible area. Let's pretend you have a piece of string, the floor is covered with Maltesers, and I'll give you the Maltesers as long as they're enclosed by your string. Assuming you like chocolate, you'd want to arrange the piece of string as much like a circle as possible to cover as many Maltesers as possible. You wouldn't want to arrange your string "like a worm" (i.e. as a long thin shape)⁠—you'd probably only get single Maltesers that fit within the worm. Imagine if you were able to make your string infinitely like a worm (i.e. maximally long and thin for that given perimeter)⁠: its area would be near zero and you wouldn't get any Maltesers at all.

The same holds true for rectangles. The closer a rectangle is to a square, the more efficient it will be at enclosing a big area⁠—in other words, the bigger its area is. Imagine the same piece of string, but instead of making circles or long thin shapes, it's making rectangles. Given the same perimeter, the longer and thinner the rectangle is, the smaller its area will be. And that's what the chart is showing you.

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Kate Allen's avatar

JCX has it nicely.

My explanation is to think of your perimeter all straightened into a single line, but made up of meccano type pieces so you can click it into all possible rectangles. Long and thin or short and fat. You might well notice that you only need half of your long perimeter to make two sides of the rectangle. You can call the longer one L and the shorter one W length and width. Or call them whatever you like. Your perimeter will always be 2 x L + 2 x W or, more elegantly, 2(L + W). Your area will always be L x W, or more elegantly LW. To make the area as big as possible you need L and W to be as close to each other in value as you can make them. Conversely, to make the area as small as possible make W (or L) as small as you can, say 1.

For example a rectangle of perimeter 20 could have its smallest area with length 9 and width 1 (area 9 squares) or its largest with length 5 and width 5 (area 25 squares).

Which of those would hold more Maltesers??

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