Difference between revisions of "Isoperimetric Inequalities"
(Changed < to \leq) |
Etmetalakret (talk | contribs) m |
||
| (5 intermediate revisions by 3 users not shown) | |||
| Line 1: | Line 1: | ||
| − | + | '''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== | ||
| Line 8: | Line 11: | ||
* [[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.
