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May be this question do not meet the standards of this blog but i was just calculating on things and i got stuck..actualy we were playing this game in which we were to fill students in a square and circle of almost same length but it was amazing to see in a square we fitted less than in a circle..thus If a square and a circle are of same length so i think they should also occupy the same area but they dont..why?

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    $\begingroup$ This is not a question of physics, but math. Also for your question to makes sense, you should give an argument why it's necessary for two objects of the same perimeter to occupy the same area. It should also be noted that circle is a geometric object with given perimeter $P$ which maximizes the area it encloses. So the area a circle encloses is bigger than that of a square,triangle or any geometric object of the same length(perimeter). Your question would be a good one, if it asked for why the circle has this property of maximizing enclosed area. Re-post it in Math stack. $\endgroup$
    – Omar Nagib
    Sep 1, 2015 at 5:42
  • $\begingroup$ You've answered your own question: you've shown area is not uniquely defined by perimeter. $\endgroup$ Sep 1, 2015 at 7:34
  • $\begingroup$ Would Mathematics be a better home for this question? $\endgroup$
    – Qmechanic
    Sep 1, 2015 at 9:46
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    $\begingroup$ I'm voting to close this question as off-topic because it is about math, and I consider it of too low quality to migrate. $\endgroup$
    – ACuriousMind
    Sep 1, 2015 at 10:46
  • $\begingroup$ This question has already been asked and answered on math SE: math.stackexchange.com/q/4808 $\endgroup$
    – Ellie
    Sep 1, 2015 at 14:44

2 Answers 2

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Imagine a rope enclosing a two-dimensional gas, with vacuum outside the rope. The gas will expand, pushing the rope to enclose a maximal area at equilibrium.

When the system is at equilibrium, the tension in the rope must be constant, because if there were a tension gradient at some point, there would be a non-zero net force at that point in the direction of the rope, but at equilibrium the net force must be zero in all directions.

The gas exerts a force outward on the rope, so tension must cancel this force. Take a small section of rope, so that it can be thought of as a part of some circle, called the osculating circle. The force on this rope segment due to pressure is Pl, with P pressure and l the length. The net force due to tension is 2Tsin(l/2R), with T tension and R the radius of the osculating circle.

Because the pressure is the same everywhere, and the force from pressure must be canceled by the force from tension, the net tension force must be the same for any rope segment of the same length. That means the radius of the osculating circle is the same everywhere, so the rope must be a circle.

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One basic reason is a matter of dimensionality $d$ (and units), topology and intuition behind "measurements". Typically, a perimeter is of dimension one (meters), a surface of dimension 2 ($m^2$), a volume of dimension 3 ($m^3$). As they do not belong to the same dimensional space, they may be unrelated. Then, one can have two $K$-dimensional objects with the same given measurement in dimension $d_{k\in K}$, and different measurements in other dimensions. Your problem with the square and the circle has a long history, and is know as Queen Dido's problem, or isoperimetric inequality, the search for surfaces with the same perimeter. As the solution is not unique in standard spaces, one may constrain the problem with other wishes, such as having the maximum surface, of the flattest, the most regular). This comes as a problem in physics or chemistry, in materials design, when one wants to maximize a contact surface with a given quantity of matter (volume), or with soap bubbles (minimal surfaces).

You can get even less inuitive shapes. Fractals like the Koch snowflake provide examples of an infinite standard dimension (eg perimeter) with a finite other dimension (area). The Menger sponge possesses infine surface, and zero volume. For this reason, the notion of fractal dimension is useful to seize how a zero-standard-dimension object can somehow fill the space.

Your question opens on a great deal of works embracing convexity, topology, regularity (of the surface). Even logic. Proper notions of measurements (area, volume), though physically intuitive, can lead to well-know mathematical paradoxes, involving the axiom of choice. Going back to your problem, mathematicians can cut a disk into a finite numbers of pieces to form a square with the same area. Do not try this at home with a scissor, it is practically impossible. And you can cut a ball of volume one and form two new balls of volume one each. This is the Banach-Tarski paradox.

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  • $\begingroup$ thanks that was useful. and you said mathematicians can cut a disk .. was that deliberate. $\endgroup$
    – usher
    Sep 2, 2015 at 15:54
  • $\begingroup$ I am not an expert on such questions. Yet, I understood areas are preserved with arbitrary cuts, while volumes are not. Of course, these cuts are non constructive, fortunately for our world, or not. $\endgroup$ Sep 2, 2015 at 16:15
  • $\begingroup$ actually my emphasis was on the word "mathematicians" $\endgroup$
    – usher
    Sep 2, 2015 at 17:02

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