A circular plate with radius 1 unit rolls around on a rectangular tray so that the edge of the plate is always in contact with the inside wall of the tray. The plate rolls around and never slides along. If the internal measurements of the tray are 3 units by 4 units, what proportion of the circumference of the plate comes into contact with the sides of the tray when it rolls around one circuit?

Does it make a difference if you change the dimensions of the tray, and if so, how?