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Geometry Difficulty 4.4 AIME Find the answer Slovenia

We have a cylinder with a radius of $1$ cm and a height of $4$ cm. While the point $P$ lies on the bottom circle of the cylinder, the point $Q$ lies on the top circle right above $P$ . A string connecting the point $P$ and $Q$ makes a complete turn round the cylinder (see the picture). How long is the shortest such string in cm?

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Solution

If we cut the cylinder along the vertical line segment $PQ$ and unfold it, we get a rectangle with sides of lengths $4$ cm and $2\pi$ cm. Now the problem asks for the shortest path between two opposite vertices of the rectangle. The length of such a path is the same as the length of the diagonal, hence $\sqrt{(2\pi)^2 + 4^2} = 2\sqrt{\pi^2 + 4}$ cm.

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Source: MathNet, licensed CC-BY-4.0. Statement reproduced verbatim; metadata (topic, difficulty) added by this project.