Discussion about math, puzzles, games and fun. Useful symbols: ÷ × ½ √ ∞ ≠ ≤ ≥ ≈ ⇒ ± ∈ Δ θ ∴ ∑ ∫ π -¹ ² ³ °

You are not logged in.

- Topics: Active | Unanswered

Pages: **1**

**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 ?!

ImPo$$!BLe = NoTH!nG

Go DowN DeeP iNTo aNyTHinG U WiLL FinD MaTHeMaTiCs ...

Offline

**ganesh****Moderator**- Registered: 2005-06-28
- Posts: 22,965

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.

It is no good to try to stop knowledge from going forward. Ignorance is never better than knowledge - Enrico Fermi.

Nothing is better than reading and gaining more and more knowledge - Stephen William Hawking.

Offline

**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)*

Support MathsIsFun.com by clicking on the banners.

What music do I listen to? Clicky click

Offline

**MathsIsFun****Administrator**- Registered: 2005-01-21
- Posts: 7,646

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

Offline

**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.

IPBLE: Increasing Performance By Lowering Expectations.

Offline

Pages: **1**