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**RauLiTo****Member**- From: Bahrain
- Registered: 2006-01-11
- Posts: 142

a triangle and a square and a circle have got the same area which one has got the biggest circumference ?

i think the triangle ... can i see your solution for that ?!

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**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 21,298

RauLiTo,

You are correct!

It is very easy to construct a scalene triangle of small area with a very large perimeter.

Lets assume your questions talks only about an equilateral triangle.

Let a² be the area of the equilateral triangle and the square.

The perimeter of the square would be 4a, where a is the length of each side a and a² the area of the square.

The formula for area of an equilateral triangle is

where s is the length of each side.

Equating the area a² and the formula for the area,

Thus, it is seen that the perimeter of an equilateral triangle is more than that os a square of same area.

As the number of sides of a regular polygon increases, the perimeter minimses to give maximum area, culminating in a circle.

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**Patrick****Real Member**- Registered: 2006-02-24
- Posts: 1,005

edit: too slow :] - but ganesh you seem to have made some errors in the LaTeX

the first one had no 'a' in frac

and is this what you meant in the second? (a _ where there should be none I think)

Same problem in the last one

*Last edited by Patrick (2006-06-09 02:08:26)*

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**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,626

A circle is the solution to smallest circumference for area, and likewise a sphere for smallest surface for volume, and that is why they are common in nature (in a non-perfect way of course).

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**krassi_holmz****Real Member**- Registered: 2005-12-02
- Posts: 1,905

A note: there may exist a figure with arbitary circumfence and constant area.

For the original question:

The figure with the biggest circumfence is the most "angular"(the most different from a circle) figure.

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