Why do squares have the largest area

Now, I pointed out, look at the blue and yellow rectangles and the dotted square in the upper right corner. What can you say about their areas. He immediately saw that.

Contact Front page Contents Did you know? What is what? Isoperimetric Theorem for Rectangles A particular case of the Isoperimetric Theorem says that among all rectangles of a given perimeter, the square has the largest area. The problem has been posed in the following form: Mr. Related material Read more Isoperimetric Theorem and Inequality. An Isoperimetric theorem. Isoperimetric theorem and its variants.

Isoperimetric Property of Equilateral Triangles. Maximum Area Property of Equilateral Triangles. Isoperimetric Theorem For Quadrilaterals. Everything has a reason for being the way it is. There was a reason the inventors of mathematics created it the way they did. A human created the formula to determine the area and the perimeter, etc of something. I see! You're asking the question "Why does the square have larger area than the rectangle?

They could have chosen otherwise, but they chose this. But here I don't think people had that much choice in the matter. Area is just a name for the number of little squares that can fit inside a given shape.

Perimeter is a name for the number of little line segments that can go along the boundary of the shape. Those would still be there even if we hadn't given those concepts a name, just as a lemon would have a colour and a taste even if we didn't have words for yellow and sour.

Go ahead, try it right now! What part of that feels invented, or man-made, or arbitrary? Did someone else choose those numbers, when you counted them yourself? It's kind of implicit in your original framing of the question, too, isn't it? And you want to know why someone defined the formulas so that it works out this way.

The idea that some mathematicians in antiquity are responsible, that if only they had defined area and perimeter differently it would somehow change the amount of stuff you can fit into your rooms, well, it doesn't make any sense to me. The second derivative test confirms that this is a maximum. Cut yourself a length of string and tie it together to make a loop. You can lay this out on the ground so that it surrounds many different sized areas.

There will be a maximum possible area. This is a famous problem known as the Isoperimetric Problem. It turns out that the maximum enclosed area is given by making a circle. But what about the smallest possible enclosed area? Well, if you use "mathematical string", i. Think about a long, thin rectangle. If you want the enclosed area to be less then make the rectangle longer and thinner. What about the maximum possible area for a rectangle?

A circle encloses the maximum possible area. This means we have made a square. TLDR : There is no general connection between the perimeter and the enclosed area. You can make an enclosed area as small as you like for any given perimeter. There is, however, a maximum possible area for any given perimeter; this is formed by a circle.

I'll try to give an intuitive answer. For the rigorous mathematical argument you have Claude's answer. You'll have two internal squares, in which two sides are belong to the perimeter, and two external squares, in which three sides belong to the perimeter. Any rectangle will always have more of these blocks exposed to the outside than a square of the same area. This proves that a rectangle will always have a larger perimeter than a square with the same area.

This implies that if a rectangle and a square have the same perimeter, the rectangle must have a smaller area. Suppose you have a rectangle that isn't a square. Then it's longer in one dimension than the other. Since you added 1 inch to two sides, and removed 1 inch from two sides, the perimeter stays the same.

Since the long sides were longer than the short sides to begin with, you added more area than you subtracted. Therefore you can always increase a rectangle's area, without changing its perimeter, by transferring length from the long sides to the short sides as long as the amount of length that you transfer isn't more than half the difference between them. The only time this isn't possible is when all of the sides are the same length — that is, when the rectangle is a square.

Therefore, a square maximizes area for a given perimeter. This is both in response to the comments made in Rahul's answer, and also as an additional graphical proof. In addition, let me emphasize that to mathematicians, "Why" something is true is explained in mathematical proof. Explaining things via formula and pictures and such is not a "How" but is indeed a "Why". Also let me emphasize that when beginning anything it is important to remember the definitions.

Claim: All rectangles of a given perimeter have less than or equal area to that of the square of the same perimeter. Equivalently, a square of a given perimeter is of greater area than all rectangles of equal perimeter where the rectangle is not a square.

Without loss of generality let the rectangle we are comparing to be longer than it is tall. By removing the upper-right piece you effectively reduce the area. You may then move the upper-left piece and rotate it to fit nicely with the lower piece. The resulting formed rectangle is of the same perimeter as the original square, however the area is smaller. If you are looking for a simpler answer, the "why" is because area and perimeter measure two completely different things.

If you are looking for a more philosophical answer, then I would direct you to read Plato and Pythagoras. Take a 13x13 square, and for convenience's sake, draw it on a 1x1 grid, so it takes up 13x13 squares. If you wanted a non-square rectangle, then the closer the sides are to equal, the bigger it would be.

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