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Prove-or-disprove-that-the-circle-has-the-largest-Perimiter-over-all-natural-shapes-that-have-area-A-

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Question Number 5417 by FilupSmith last updated on 14/May/16
Prove, or disprove, that the circle has the largest Perimiter over all natural shapes that have area A
$$\mathrm{Prove},\:\mathrm{or}\:\mathrm{disprove},\:\mathrm{that}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{has}\:\mathrm{the}\:\mathrm{largest}\:{Perimiter}\:\mathrm{over}\:\mathrm{all} \\ $$$$\mathrm{natural}\:\mathrm{shapes}\:\mathrm{that}\:\mathrm{have}\:\mathrm{area}\:{A} \\ $$
Commented by Rasheed Soomro last updated on 14/May/16
I think that the circle has the smallest perimeter among all closed figures having same area A. Let′s compare the perimeters of a circle and a square who have same area A. π r^2 =s^2 ⇒s=(√π) r (r is radius of the circle and s is measure of the side of the square.) perimeter of circle(circumference)=2πr perimeter of square=4s=4(√π) r 2πr ? 4(√π) r 2π ? 4(√π) 6.283 <7.09 circumference <perimeter of square. This disproves the claim.
$$\mathrm{I}\:\mathrm{think}\:\mathrm{that}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{has}\:\mathrm{the}\:\mathrm{smallest} \\ $$$$\mathrm{perimeter}\:\mathrm{among}\:\mathrm{all}\:\mathrm{closed}\:\mathrm{figures} \\ $$$$\mathrm{having}\:\mathrm{same}\:\mathrm{area}\:\mathrm{A}. \\ $$$$\mathrm{Let}'\mathrm{s}\:\mathrm{compare}\:\mathrm{the}\:\mathrm{perimeters}\:\mathrm{of}\:\mathrm{a}\:\mathrm{circle} \\ $$$$\mathrm{and}\:\mathrm{a}\:\mathrm{square}\:\mathrm{who}\:\mathrm{have}\:\mathrm{same}\:\mathrm{area}\:\mathrm{A}. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\pi\:\mathrm{r}^{\mathrm{2}} =\mathrm{s}^{\mathrm{2}} \Rightarrow\mathrm{s}=\sqrt{\pi}\:\mathrm{r} \\ $$$$\left(\mathrm{r}\:\mathrm{is}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle}\:\mathrm{and}\:\mathrm{s}\:\mathrm{is}\:\mathrm{measure}\right. \\ $$$$\left.\mathrm{of}\:\mathrm{the}\:\mathrm{side}\:\mathrm{of}\:\mathrm{the}\:\mathrm{square}.\right) \\ $$$$\:\:\mathrm{perimeter}\:\mathrm{of}\:\mathrm{circle}\left(\mathrm{circumference}\right)=\mathrm{2}\pi\mathrm{r} \\ $$$$\mathrm{perimeter}\:\mathrm{of}\:\mathrm{square}=\mathrm{4s}=\mathrm{4}\sqrt{\pi}\:\mathrm{r} \\ $$$$\:\:\:\:\mathrm{2}\pi\mathrm{r}\:\:\:?\:\:\mathrm{4}\sqrt{\pi}\:\mathrm{r} \\ $$$$\:\:\:\:\:\mathrm{2}\pi\:\:\:\:\:?\:\:\:\mathrm{4}\sqrt{\pi} \\ $$$$\:\:\:\:\:\mathrm{6}.\mathrm{283}\:\:<\mathrm{7}.\mathrm{09} \\ $$$$\:\:\:\:\:\mathrm{circumference}\:<\mathrm{perimeter}\:\mathrm{of}\:\mathrm{square}. \\ $$$$\mathrm{This}\:\mathrm{disproves}\:\mathrm{the}\:\mathrm{claim}. \\ $$
Commented by Rasheed Soomro last updated on 14/May/16
The claim should be for ′smallest perimeter′ instead of ′largest perimeter′.
$$\mathrm{The}\:\mathrm{claim}\:\mathrm{should}\:\mathrm{be}\:\mathrm{for}\:'\mathrm{smallest} \\ $$$$\mathrm{perimeter}'\:\mathrm{instead}\:\mathrm{of}\:'\mathrm{largest}\:\mathrm{perimeter}'. \\ $$

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