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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?

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  1. Brian Gladman permalink


    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 \(2\).

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