What is the parallel axis theorem and how is it used? Please explain the equation.

Explanation:

If an object has a moment of inertia I₀ about an axis, then the moment of inertia about a different, parallel axis is I = I₀ + md², where d is the distance between the axes.

For example, consider a horizontal thin rod rotating about a vertical axis passing through its center.  It has mass m and length L.  Its moment of inertia is known to be I = 1/12 mL².

Now consider the same rod, but this time we move the axis of rotation L/2 to the end of the rod.  We can use parallel axis theorem to find the new moment of inertia:

I = I₀ + md²

I = 1/12 mL² + m (L/2)²

I = 1/12 mL² + 1/4 mL²

I = 1/3 mL²

mass and volume are the answers you are looking for