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