Let PQRS be the rectangle inscribed in a given circle with centre O and radius
Let and be the length and breadth of the rectangle, i.e., and
In right angled triangle PQR, using Pythagoras theorem,
PQ2 + QR2 = PR2
…..(i)
Let A be the area of the rectangle, then A = =
=
And
=
=
Now
= 0
At , [Negative]
At , area of rectangle is maximum.
And from eq. (i), ,
i.e.,
Therefore, the area of inscribed rectangle is maximum when it is square.