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At the corner of the cube circular arcs are drawn and the area enclosed shaded. What fraction of the surface area of the cube is shaded? Try working out the answer without recourse to pencil and paper.

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Diagonal Touch

Age 14 to 16 Short Challenge Level:


$\frac{16}{81}$ is shaded.

Let $x$ and $y$ be the distances shown. Then the shaded area is $8y + x$. But there are a number of similar triangles and from one pair $${x\over8} = {y\over1}$$ i.e. $$x = 8y$$ So, $$\frac{\mbox{shaded area}}{\mbox{total area}} = \frac{8y + x}{9(x + y)} = \frac{8y + 8y}{9 \times 9y} = \frac{16}{81}$$
This problem is taken from the UKMT Mathematical Challenges.
You can find more short problems, arranged by curriculum topic, in our short problems collection.