Math Is Fun Forum

RauLiTo

Member

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 ?!

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Jai Ganesh

Administrator

Registered: 2005-06-28

Posts: 48,429

Re: the biggest circumference !?

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

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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MathsIsFun

Administrator

Registered: 2005-01-21

Posts: 7,713

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).

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krassi_holmz

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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