# 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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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: $r_1=\frac{\sqrt{2}-1}{2}$
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 $$2$$.