If you have a regular dodecahedron and you slice it in half along a plane that passes through the centres of opposite faces, what is the shape of the cross - section?

Given:

A regular dodecahedron is a Platonic solid with 12 faces, each of which is a regular pentagon. It has 20 vertices and 30 edges.

When a regular dodecahedron is sliced in half along a plane that passes through the centers of opposite faces, the cross-section will cut through 10 of its edges and 10 vertices.

Specifically, a regular dodecahedron has 6 axes of 5-fold rotational symmetry, each passing through the centers of two opposite faces. When you slice it along such a plane, the cutting plane intersects the 5 edges of one pentagonal face and the 5 edges of the opposite pentagonal face. This creates a cross-section with 10 sides.

Thus, the shape of the cross-section is a decagon.

Hence, the correct answer is "Option (1)".