Maximal Area in Rectangles with the Same Perimeter

⁠⁠⁠⁠⁠⁠⁠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 the 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 the rectangle.

- Then,

   A (x) = xy

   A (x) = x (24 - x)

   A (x) = 24x - x²

-  A' (x) = 24 - 2x

   A' (x) ⇒ 24 - 2x = 0

   A' (x) ⇒ x = 12

-  A'' (x) = – 2

    A'' (12) = – 2, which is negative.

- Therefore, area is maximum when x = 12.

- So, y = x = 24 – 12 = 12

- i.e., x = y = 12.

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