The circumference of the circle is $2\pi$. This is the distance its
centre moves each time the circle rolls for one revolution. When
the circle moves from one corner to an adjacent corner, its centre
moves a distance 2, so the circle makes $1/\pi$ revolutions. As it
needs to do this four times before the circle returns to its
original position, the number of revolutions is $4/\pi$.
This problem is taken from the UKMT Mathematical Challenges.