Q. 245.0( 3 Votes )

# Show that of all the rectangles of given area, the square has the smallest perimeter.

Answer :

Let us Consider a rectangle with length is xcm and breadth is y cm.

Since, Area of rectangle = length × breadth

A=xy …(i)

And, Perimeter of rectangle is , = 2(length + breadth)

P=2(x + y) …(ii)

Now, From (i),

Putting the value of y in equation (ii), we get

Now, differentiate P, w.r.t x , we get

If then

x^{2}=A

If

Since,

Therefore,

Therefore, A rectangle with given area will have atleast perimeter when x=y or it is a square.

**Hence, Proved**

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