Difference between revisions of "Isoperimetric Inequalities"

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===Isoperimetric Inequality===
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'''Isoperimetric Inequalities''' are [[inequalities]] concerning the [[area]] of a figure with a given [[perimeter]]. They were worked on extensively by [[Lagrange]].
If a figure in the plane has area <math>A</math> and perimeter <math>P</math> then <math>\frac{4\pi A}{P^2} \leq 1</math>. This means that given a perimeter <math>P</math> for a plane figure, the circle has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
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If a figure in a plane has area <math>A</math> and perimeter <math>P</math> then <math>\frac{4\pi A}{P^2} \leq 1</math>. This means that given a perimeter <math>P</math> for a plane figure, the circle has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
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Note that due to this inequality, it is impossible to have a figure with infinite volume yet finite surface area.
  
 
==See also==
 
==See also==
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* [[Area]]
 
* [[Area]]
 
* [[Perimeter]]
 
* [[Perimeter]]
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[[Category:Geometry]]
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[[Category:Inequality]]
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[[Category:Theorems]]

Latest revision as of 13:07, 11 June 2008

Isoperimetric Inequalities are inequalities concerning the area of a figure with a given perimeter. They were worked on extensively by Lagrange.

If a figure in a plane has area $A$ and perimeter $P$ then $\frac{4\pi A}{P^2} \leq 1$. This means that given a perimeter $P$ for a plane figure, the circle has the largest area. Conversely, of all plane figures with area $A$, the circle has the least perimeter.

Note that due to this inequality, it is impossible to have a figure with infinite volume yet finite surface area.

See also

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