Rectangle is inscribed in a circle with radius .

 and  are the side of the rectangle.

 is the radius of the circle.

Observe the figure.

From pythagorean theorem, .

The area of the recatngle is .

The area of the rectangle is maximum when its first derivative becomes zero.

Apply derivative on each side with respect to .

.

Equate  to zero.

The length of the rectangle is .

Substitute  in .

The width of the rectangle is .

Here length and width of the rectangle is equal, so it is a square.

Length and width of the rectangle is , so it is a square.