need a deravation for parallel axis theorem

In physics, the **parallel axis theorem** or **Huygens–Steiner theorem** can be used to determine the second moment of area or the mass moment of inertia of a rigid body about any axis, given the body's moment of inertia about a parallel axis through the object's centre of mass and the perpendicular distance (* r*) between the axes.

The moment of inertia about the new axis *z* is given by:

where:

- is the moment of inertia of the object about an axis passing through its centre of mass;
- is the object's mass;
- is the perpendicular distance between the two axes.

This rule can be applied with the stretch rule and perpendicular axis theorem to find moments of inertia for a variety of shapes.

The parallel axes rule also applies to the second moment of area (area moment of inertia) for a plane region *D*:

where: