[ Cylinder ]  [

The radius of the base and the height of the cylinder are equal to $r$ and $h$ respectively. Find the length of the shortest path along the lateral surface of the cylinder between diametrically opposite points on different bases.

#
Let rectangle $A B C D$ be the development of the lateral surface of a cylinder (Fig.2), with $A B=h, A D=2 \pi r$. The desired path is equal to the shortest path between point $A$ and the midpoint $M$ of side $B C$, i.e., the length of segment $A M$. By the Pythagorean theorem in the right triangle $A B M$, we find that

$$
A M=\sqrt{B M^{2}+A B^{2}}=\sqrt{\pi^{2} r^{2}+h^{2}}
$$

## Answer

$\sqrt{\pi^{2} r^{2}+h^{2}}$.