3 [ Shortest path on the surface ]

A sphere of radius 2 is intersected by a plane at a distance of 1 from the center. Find the length of the shortest path on the surface of the sphere between the two most distant points of the intersection.

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The shortest path on a sphere connecting two of its points $A$ and $B$ is the smaller of the two arcs $AB$ of the great circle passing through points $A$ and $B$. The section of the sphere by a plane is a circle. The most distant points of this circle are any two diametrically opposite points $A$ and $B$. Through points $A$

, $B$ and the center $O$ of the sphere, we draw a plane. In the section of the sphere by this plane, we obtain a circle of radius 2.

The chord $AB$ of this circle is at a distance of 1 from its center $O$. Therefore, the smaller arc $AB$ of this circle is seen from point $O$ at an angle of $120^{\circ}$. Hence, the length of this arc is one-third of the circumference of a circle of radius 2, i.e., $\frac{1 \pi}{3}$.

## Answer

$\frac{\pi}{3}$