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 ?!
ImPo$$!BLe = NoTH!nG
Go DowN DeeP iNTo aNyTHinG U WiLL FinD MaTHeMaTiCs ...
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
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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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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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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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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