The moment of inertia is minimal when the rotation axis passes through the center-of-mass and increases as the rotation axis is moved further from the center-of-mass.

I = I_cm + Mh²

Here M is the mass, h is the distance from the center-of-mass to the parallel axis of rotation, and I_cm is the moment of inertia about the center of mass parallel to the current axis.

Example:

What is the moment of inertia I_cm for a uniform rod of length L and mass M rotating about an axis through the center, perpendicular to the rod?

I = I_cm + Mh²   →   I_cm = I - Mh²

Now the moment of inertia about the end of the rod is:   I = ML²/3.

The distance from the end of the rod to the center is h = L/2. Therefore:

Icm

=

1 3

ML2

-

M

(

L 2

)

2

=

1 3

ML2

-

1 4

ML2

=

1 12

ML2