# Sunday Times Teaser 2552 – Ever-Increasing Circles

*by Andrew Skidmore*

#### Published: 21 August 2011 (link)

I took a piece of A4 paper and cut it straight across to make a square and a rectangle. I then cut the square along a diagonal and, from the rectangle, cut as large a circle as I could. My friend Ron said that if I needed two triangles of those sizes and as large a circle in one piece as possible from an A4 sheet, then he could do much better. With a fresh sheet of A4, he produced the two triangles and the biggest circle possible.

What is the area of his circle divided by the area of mine?

One Comment
Leave one →

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 \(2\).