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== | ||
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[[Category:Geometry]] | [[Category:Geometry]] | ||
| − | [[Category: | + | [[Category:Inequalities]] |
| − | [[Category: | + | [[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.
