Difference between revisions of "Isoperimetric Inequalities"
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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]]. | |
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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. | 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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* [[Perimeter]] | * [[Perimeter]] | ||
| + | [[Category:Geometry]] | ||
| + | [[Category:Inequality]] | ||
[[Category:Theorems]] | [[Category:Theorems]] | ||
Revision as of 12:58, 22 December 2007
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 
and perimeter 
then 
. This means that given a perimeter for a plane figure, the circle has the largest area. Conversely, of all plane figures with area , the circle has the least perimeter.
