Folklore

A narrow ribbon is wound around a cylindrical column that is 20 meters high and 3 meters in diameter, rising from the base to the top in seven full turns. What is the length of the ribbon?
By cutting the cylinder along the generatrix of its lateral surface passing through the beginning of the tape and unfolding this surface onto a plane, we obtain a rectangle $A B C D$ with dimensions $20 \times 3 \pi$ (see figure). By "gluing" six more such rectangles to it, we get rectangle $A B Q R$.

![](https://cdn.mathpix.com/cropped/2024_05_06_dc8ed1d6cdac2bf77445g-37.jpg?height=414&width=1144&top_left_y=1371&top_left_x=457)

On this development, the first turn of the tape will appear as segment $A P$, and the entire tape will unfold into the diagonal $A Q$ of rectangle $A B Q R$. Therefore, its length $l = A Q = \sqrt{R Q^{2} + A R^{2}} = \sqrt{20^{2} + (21 \pi)^{2}} = \sqrt{400 + 441 \pi^{2}} \approx 68.9$.

## Answer

$$
\sqrt{400 + 441 \pi^{2}} \text{ m. }
$$

Submit a comment