Moment of inertia, also known as the rotational inertia, is a property of a body that determines its resistance to rotational motion about a particular axis. It depends on the mass distribution of the body and the axis of rotation.

The moment of inertia of a square about an axis passing through its center of gravity can be calculated using the parallel axis theorem. According to this theorem, the moment of inertia about an axis parallel to and a distance 'd' away from an axis passing through the center of gravity is given by the sum of the moment of inertia about the center of gravity and the product of the mass of the body and the square of the distance 'd'.

In the case of a square of side 'b', the moment of inertia about an axis passing through its center of gravity is given by:

I = Icg + mb²

Where:
I is the moment of inertia about the axis passing through the center of gravity,
Icg is the moment of inertia about the center of gravity, and
m is the mass of the square.

The moment of inertia about the center of gravity of a square can be calculated using the formula:

Icg = (1/12)mb²

Substituting this into the equation above, we get:

I = (1/12)mb² + mb²
I = (1/12 + 1)mb²
I = (13/12)mb²

Simplifying further, we get:

I = (13/12)mb² = (b²/12)(13m)
I = (b⁴/12)(13/12)
I = b⁴/12

Therefore, the moment of inertia of a square of side 'b' about an axis passing through its center of gravity is given by b⁴/12, which corresponds to option 'C'.