# Difference between revisions of "Isoperimetric Inequalities"

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

==See also== | ==See also== |

## 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 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.

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