Isoperimetric Inequalities: Difference between revisions
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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 | |||
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. | |||
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]] | ||
[[Category:Geometry]] | |||
[[Category:Inequalities]] | |||
[[Category:Geometric Inequalities]] | |||
Latest revision as of 16:08, 29 December 2021
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.
