The moment of inertia with respect to a given axis of a solid body with density 
is defined by the volume integral
 |
(1)
|
where 
is the perpendicular distance from the axis of rotation. This can be broken into
components as
 |
(2)
|
for a discrete distribution of mass, where 
is the distance to a point (not the perpendicular distance)
and 
is the Kronecker delta, or
 |
(3)
|
for a continuous mass distribution. Depending on the context, 
may be viewed either as a tensor or a matrix. Expanding (3) in terms of Cartesian axes gives the equation
![I=int_Vrho(x,y,z)[y^2+z^2 -xy -xz; -xy z^2+x^2 -yz; -xz -yz x^2+y^2]dxdydz.](/images/equations/MomentofInertia/NumberedEquation4.svg) |
(4)
|
The moment of inertia of a region can be computed in the Wolfram Language using MomentOfInertia[reg].
The moment of inertia tensor 
is symmetric, and is related to the angular momentum vector
by
 |
(5)
|
where 
is the angular velocity vector.
The principal moments of inertia are given by the entries in the diagonalized moment of inertia matrix, and are denoted (for a solid) 
, 
, and 
in order of decreasing magnitude. In the principal axes frame,
the moments are also sometimes denoted 
, 
, and 
. The principal axes of a rotating body are defined by
finding values of 
such that
![L=[L_x; L_y; L_z]=[I_(11) I_(12) I_(13); I_(21) I_(22) I_(23); I_(31) I_(32) I_(33)][omega_x; omega_y; omega_z]=I[omega_x; omega_y; omega_z]](/images/equations/MomentofInertia/NumberedEquation6.svg) |
(6)
|
![[I_(11)-I I_(12) I_(13); I_(21) I_(22)-I I_(23); I_(31) I_(32) I_(33)-I][omega_x; omega_y; omega_z]=[0; 0; 0],](/images/equations/MomentofInertia/NumberedEquation7.svg) |
(7)
|
which is an eigenvalue problem.
The following table summarizes the moments of inertia of some common solids around some of their principal axes.
| solid/axis | moment of inertia |
| cylinder about symmetry axis |  |
| cylinder about central diameter |  |
| ellipsoid about principal axis |  |
| elliptical slab about major axis |  |
| elliptical slab about vertical |  |
| cuboid about major axis |  |
| ring about perpendicular axis |  |
| ring about diameter |  |
| rod about end |  |
| rod about center |  |
| sphere about diameter |  |
| spherical shell |  |
| torus about diameter |  |
| torus about symmetry axis |  |
