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Question Number 3417 by RasheedSindhi last updated on 13/Dec/15
prove that among all closed figures  having same perimeter circle  has maximum area.
$${prove}\:{that}\:{among}\:{all}\:{closed}\:{figures} \\ $$$${having}\:{same}\:{perimeter}\:{circle} \\ $$$${has}\:{maximum}\:{area}. \\ $$
Commented by Filup last updated on 13/Dec/15
I dont understand the question
$$\mathrm{I}\:\mathrm{dont}\:\mathrm{understand}\:\mathrm{the}\:\mathrm{question} \\ $$
Commented by RasheedSindhi last updated on 13/Dec/15
Consider closed figures triangle,  quadilateral,pentagon...or other  closed curves or mixture of curves  and straight line segments ,all  having same perimeter.Among  all the areas these figures have  the largest is the area of circle.
$${Consider}\:{closed}\:{figures}\:{triangle}, \\ $$$${quadilateral},{pentagon}…{or}\:{other} \\ $$$${closed}\:{curves}\:{or}\:{mixture}\:{of}\:{curves} \\ $$$${and}\:{straight}\:{line}\:{segments}\:,{all} \\ $$$${having}\:{same}\:{perimeter}.{Among} \\ $$$${all}\:{the}\:{areas}\:{these}\:{figures}\:{have} \\ $$$${the}\:{largest}\:{is}\:{the}\:{area}\:{of}\:{circle}. \\ $$
Commented by 123456 last updated on 13/Dec/15
this is isoperemtric inequality  4πA≤L^2
$$\mathrm{this}\:\mathrm{is}\:\mathrm{isoperemtric}\:\mathrm{inequality} \\ $$$$\mathrm{4}\pi\mathrm{A}\leqslant\mathrm{L}^{\mathrm{2}} \\ $$
Commented by Filup last updated on 13/Dec/15
Circle is essentially an ∞−sided  polygon
$$\mathrm{Circle}\:\mathrm{is}\:\mathrm{essentially}\:\mathrm{an}\:\infty−{sided} \\ $$$${polygon} \\ $$
Commented by Filup last updated on 13/Dec/15
3−gon (triangle)<4−gon (square)  4−gon<5−gon  etc  ∴n−gon<∞−gon (circle)
$$\mathrm{3}−\mathrm{gon}\:\left(\mathrm{triangle}\right)<\mathrm{4}−\mathrm{gon}\:\left(\mathrm{square}\right) \\ $$$$\mathrm{4}−\mathrm{gon}<\mathrm{5}−\mathrm{gon} \\ $$$$\mathrm{etc} \\ $$$$\therefore{n}−{gon}<\infty−{gon}\:\left({circle}\right) \\ $$
Commented by Rasheed Soomro last updated on 13/Dec/15
What is meant by isoperemtric inequality ?
$$\mathcal{W}{hat}\:{is}\:{meant}\:{by}\:\mathrm{isoperemtric}\:\mathrm{inequality}\:? \\ $$
Commented by RasheedSindhi last updated on 13/Dec/15
Th∝nkS
$$\mathcal{T}{h}\propto{nk}\mathcal{S} \\ $$
Commented by prakash jain last updated on 13/Dec/15
isoperimetric inequality for all closed  plane figure  4πA≤L^2   A area  L perimeter  equality holding true only for circle.
$$\mathrm{isoperimetric}\:\mathrm{inequality}\:\mathrm{for}\:\mathrm{all}\:\mathrm{closed} \\ $$$$\mathrm{plane}\:\mathrm{figure} \\ $$$$\mathrm{4}\pi\mathrm{A}\leqslant\mathrm{L}^{\mathrm{2}} \\ $$$$\mathrm{A}\:\mathrm{area} \\ $$$$\mathrm{L}\:\mathrm{perimeter} \\ $$$$\mathrm{equality}\:\mathrm{holding}\:\mathrm{true}\:\mathrm{only}\:\mathrm{for}\:\mathrm{circle}. \\ $$

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