Measuring the lengths around the circumference of the plate we refer to 'arc points' on the circumference. Now the arc from approximately 3.571 units to 4.571 units around the circumference comes into contact with the second edge of the tray.

Then at the next corner an arc of length $\frac{\pi }{2}\approx 1.571$ does not touch the tray so the next 'arc point' to come in contact is approximately 6.142 units around the circumference.

Continuing to roll on the third edge the points from 'arc point' 6.142 to 'arc point' 8.142 are in contact with the tray. Note that 'arc point' $2\pi \approx 6.283$ is the point on the plate which was in contact with the tray at the start and we shall need to find which arc lengths beyond this point make contact as the plate continues to roll to complete the first circuit of the tray.

Finally, taking the measurements correct to 3 decimal places, on the fourth edge the points from 8.142 + 1.571 = 9.713 to 'arc point' 10.713 are in contact and this completes one circuit.

If we started measuring again from 'arc point' 6.283 (equivalently arc point 0) then the arcs from 0 to 8.142-6.283=1.859 and from 3.430 to 4.430 make contact.

The total length of arc of the plate that makes contact is 2 + 1 + (6.283-6.142) on the first revolution of the plate plus (3.571 - 3.430) on the second revolution making a total arc length of 3.282.

So the fraction of the circumference of the plate which comes into contact with the tray on the first circuit is approximately $\frac{3.282}{2\pi }=0.52$ to 2 significant figures.