# derivation of formula of parallel and prependicular axis theorem

Parallel Axis Theorem, Moment of Inertia Proof

The theorem determines the moment of inertia of a rigid body about any given axis, given that moment of inertia about the parallel axis through the center of mass of an object and the perpendicular distance between the axes.

Proof:

Assume that the perpendicular distance between the axes lies along the x-axis and the centre of mass lies at the origin. The moment of inertia relative to z-axis that passes through the centre of mass, is represented as

Moment of inertia relative to the new axis with its perpendicular distance r along the x-axis, is represented as:

We get,

The first term is Icm,the second term is mr2and the final term is zero as the origin lies at the centre of mass. Finally,