Topic: Rigid Bodies
Concepts: Moments of inertia, Parallel-axis theorem

Solution:

We are given that the moment of inertia $I_1$ of an equilateral triangle (axis through side) is

Let’s first find the moment of inertia $I_2$ of an equilateral triangle (axis through vertex). We use the parallel axis theorem:

Subtracting these two equations,

The hexagon can be broken down into 6 smaller equilateral triangles. Thus, the moment of inertia about an axis through opposite vertices of a hexagon is equivalent to the moment of inertia of 4 triangles (axis through side) and 2 triangles (axis through vertex). If $m$ is the mass of the hexagon, then each triangle has mass $m/6$ so

Hence, the answer is B.