Objective

To verify that amongst all the rectangles of the same perimeter, the square has the maximum area.

Mathematical example

Let the length and breadth of rectangle be x and y.

The perimeter of the rectangle P = 48 cm.

2 (x + y) = 48

or

x + y=24 or y=24-x

Let A (x) be the area of rectangle, then

A (x) = xy

= x (24 - x)

= 24x - x2

A′ (x) = 24 – 2x

A′ (x) = ⇒ 24 – 2x = 0 ⇒ x = 12

A′′ (x) = – 2

A′′ (12) = – 2, which is negative

Therefore, area is maximum when x = 12

y = x = 24 – 12 = 12

So, x = y = 12

Hence, amongst all rectangles, the square has the maximum area.