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#1 2006-06-08 20:08:45

From: Bahrain
Registered: 2006-01-11
Posts: 142

the biggest circumference !?

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 ?! rolleyes

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#2 2006-06-09 01:58:31

Registered: 2005-06-28
Posts: 20,762

Re: the biggest circumference !?

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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#3 2006-06-09 01:59:03

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

Re: the biggest circumference !?

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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#4 2006-06-09 11:51:39

Registered: 2005-01-21
Posts: 7,608

Re: the biggest circumference !?

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).

"The physicists defer only to mathematicians, and the mathematicians defer only to God ..."  - Leon M. Lederman


#5 2006-06-10 02:54:42

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

Re: the biggest circumference !?

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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