When the disc is in the corner there is one quarter of the circumference (that is a length 2p/4 = p/2) which does not come in contact with the edge of the tray. As the disc rolls round one circuit: an arc of length 2 units comes into contact, then an arc of length p/2 gets missed, then an arc of length 1 unit comes into contact, then an arc of length pi/2 gets missed, then an arc of length 2 units comes into contact, then an arc of length p/2 gets missed, then an arc of length 1 unit comes into contact.

As an extension to this problem, can you now work out how much of the circumference of the disc never touches the edge of the tray?

The article 'A Rolling Disc' discusses variations of this problem.