Math Questions and Answers

Start Your Free Trial

Among all rectangles of given area, show that the square has the least perimeter.

Expert Answers info

Eric Bizzell eNotes educator | Certified Educator

briefcaseTeacher (K-12)

calendarEducator since 2011

write3,177 answers

starTop subjects are Math, Science, and Business

Let the sides of the rectangle be given by x and y. Then the area is `A=xy` and the perimeter is `P=2x+2y` .

`A` is a given constant. Then `x=A/y` .


We wish to minimize the function, so take the first derivative and find the extrema:

`(dP)/(dy)=(-2A)/y^2+2` The critical points occur when the derivative doesn't exist, or when the derivative is zero. The derivative fails if y=0, but this doesn't make sense for the problem. Then setting the derivative equal to zero we get:




Thus the minimum occurs when `y=sqrt(A)` . Since `x=A/y` we have `x=A/sqrt(A)=sqrt(A)` so x=y and the rectangle must be a square.

check Approved by eNotes Editorial