The length and width of an A4 page are in the ratio of \(1\) to \(\sqrt{2}\) as shown on the left.

The radius of the largest cicle in the remaining area when the largest possible square is cut from the A4 page is easily shown to be: \[r_1=\frac{\sqrt{2}-1}{2}\]

To obtain a larger circle, the second triangle is moved so that its hypoteneuse is on a long edge of the A4 page. This leaves a kite shaped area as shown above.

The angle \(\alpha\) can be seen by inspection to be \(\pi/8\) radians and, using the addition rule for tangents, we can show that:\[\tan{(\pi/4)}=1= \frac{2\tan(\pi/8)}{1-\tan^2(\pi/8)}\]which can be solved to show that \(\tan{(\pi/8)}=\sqrt{2}-1\).

From the diagram it can be seen that:\[r_2+r_2/\tan{(\pi/8)}=1\] from which the value of \(r_2\) can be derived using the value of \(\tan{(\pi/8)}\) given above:\[r_2=\frac{\sqrt{2}-1}{\sqrt{2}}\]

Hence the ratio of the radii of the two circles is \(\sqrt{2}\), giving their area ratio as \(2\).