Result

Parallel axis theorem

I = I_{c m} + M d^{2}

Statement

The moment of inertia about any axis equals the moment about a parallel axis through the centre of mass, plus Md², where d is the distance between the axes.

Symbols and units

$I$	moment about the new axis	kg·m²
$I_{c m}$	moment about the centre-of-mass axis	kg·m²
$d$	distance between the axes	m
$M$	total mass of the object	kg

When it applies

The two axes must be parallel; d is the distance between them.

Don't use when

Only relates moments of inertia about two parallel axes — does not apply between axes at an angle to each other.

Derivation

A standard result of expanding r² about the centre-of-mass axis; the cross term vanishes because it is weighted by the (zero) position of the centre of mass relative to itself.

I = I_{c m} + M d^{2}

Worked examples

Rod rotated about its end

A uniform rod has I_{cm} = ⅓ML² about its centre (a standard result); about one end, d = L/2:

I_{end} = \frac{1}{12} M L^{2} + M (\frac{L}{2})^{2} = \frac{1}{3} M L^{2}

Watch out for

The centre-of-mass moment I_cm is always the smallest moment of inertia among all parallel axes — every other parallel axis adds a positive Md².

Connects to

Moment of inertia Centre of mass