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

Question Number 5417 by FilupSmith last updated on 14/May/16

Prove, or disprove, that the circle

has the largest P e r i m i t e r over all

natural shapes that have 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} \Rightarrow s = \sqrt{π} r

(r is radius of the circle and s is measure

of the side of the square .)

perimeter of circle (circumference) = 2 π r

perimeter of square = 4 s = 4 \sqrt{π} r

2 π r ? 4 \sqrt{π} r

2 π ? 4 \sqrt{π}

6.283 < 7.09

circumference < perimeter of square .

This disproves the claim .

Commented by Rasheed Soomro last updated on 14/May/16

The claim should be for^{'} smallest

{perimeter}^{'} instead of^{'} largest {perimeter}^{'} .