List of moments of inertia

Template:Short description The moment of inertia, denoted by Template:Mvar, measures the extent to which an object resists rotational acceleration about a particular axis; it is the rotational analogue to mass (which determines an object's resistance to linear acceleration). The moments of inertia of a mass have units of dimension ML² ([mass] × [length]²). It should not be confused with the second moment of area, which has units of dimension L⁴ ([length]⁴) and is used in beam calculations. The mass moment of inertia is often also known as the rotational inertia or sometimes as the angular mass.

For simple objects with geometric symmetry, one can often determine the moment of inertia in an exact closed-form expression. Typically this occurs when the mass density is constant, but in some cases, the density can vary throughout the object as well. In general, it may not be straightforward to symbolically express the moment of inertia of shapes with more complicated mass distributions and lacking symmetry. In calculating moments of inertia, it is useful to remember that it is an additive function and exploit the parallel axis and the perpendicular axis theorems.

This article considers mainly symmetric mass distributions, with constant density throughout the object, and the axis of rotation is taken to be through the center of mass unless otherwise specified.

Moments of inertia

The following are scalar moments of inertia. In general, the moment of inertia is a tensor; see below.

Description	Figure	Moment(s) of inertia	Notes
Point mass M at a distance r from the axis of rotation.	File:PointInertia.svg	$I=M{r}^2$	A point mass does not have a moment of inertia around its own axis, but using the parallel axis theorem a moment of inertia around a distant axis of rotation is achieved.
Two point masses, m1 and m2, with reduced mass μ and separated by a distance x, about an axis passing through the center of mass of the system and perpendicular to the line joining the two particles.	File:2PointInertia.svg	$I=\frac{m_1{m}_2}{m_1+m_2}{x}^2=\mu x^2$	Both bodies are treated as point masses: dots of different size indicate the difference in masses of bodies, not in their sizes.
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about its center.	File:Moment of inertia rod center.svg	$I_{\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}}=\frac{1}{12}m{L}^2$	This expression assumes that the rod is an infinitely thin (but rigid) wire. This is a special case of the thin rectangular plate with axis of rotation at the center of the plate, with w = L and h = 0.
Thin rod of length L and mass m, perpendicular to the axis of rotation, rotating about one end.	File:Moment of inertia rod end.svg	$I_{\mathrm{e}\mathrm{n}\mathrm{d}}=\frac{1}{3}m{L}^2$	This expression assumes that the rod is an infinitely thin (but rigid) wire. This is also a special case of the thin rectangular plate with axis of rotation at the end of the plate, with h = L and w = 0.
Thin circular loop of radius r and mass m.	File:Moment of inertia hoop.svg	$I_z=m{r}^2$ $I_x=I_y=\frac{1}{2}m{r}^2$	This is a special case of a torus for a = 0 (see below), as well as of a thick-walled cylindrical tube with open ends, with r1 = r2 and h = 0
Thin, solid disk of radius r and mass m.	File:Moment of inertia disc.svg	$I_z=\frac{1}{2}m{r}^2$ $I_x=I_y=\frac{1}{4}m{r}^2$	This is a special case of the solid cylinder, with h = 0. That $I_x=I_y=\frac{I_z}{2}$ is a consequence of the perpendicular axis theorem.
A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2	File:Moment of inertia annulus.svg	$I_z=\frac{1}{2}m(r_1^2+r_2^2)$ $I_x=I_y=\frac{1}{4}m(r_1^2+r_2^2)$
An annulus with a constant area density ${\rho }_A$		$I_z=\frac{1}{2}\pi {\rho }_A(r_2^4-r_1^4)$ $I_x=I_y=\frac{1}{4}\pi {\rho }_A(r_2^4-r_1^4)$
Thin cylindrical shell with open ends, of radius r and mass m.	File:Moment of inertia thin cylinder.png	$I=m{r}^2$	The expression ″thin″ indicates that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube of the same mass for r1 = r2.
Solid cylinder of radius r, height h and mass m.	File:Moment of inertia solid cylinder.svg	$I_z=\frac{1}{2}m{r}^2$   $I_x=I_y=\frac{1}{12}m(3r^2+h^2)$	This is a special case of the thick-walled cylindrical tube, with r1 = 0.
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.	File:Moment of inertia thick cylinder h.svg	$I_z=\frac{1}{2}m(r_2^2+r_1^2)=m{r}_2^2(1-t+\frac{t^2}{2})$   where $t=\frac{r_2-r_1}{r_2}$ is a normalized thickness ratio; $I_x=I_y=\frac{1}{12}m(3(r_2^2+r_1^2)+h^2)$	The given formula is for the $xy$ plane passing through the center of mass, which coincides with the geometric center of the cylinder. If the xy plane is at the base of the cylinder, i.e. offset by $d=\frac{h}{2},$ then by the parallel axis theorem the following formula applies: $I_x=I_y=\frac{1}{12}m(3(r_2^2+r_1^2)+4h^2)$
With a density of ρ and the same geometry		$I_z=\frac{\pi \rho h}{2}(r_2^4-r_1^4)$ $I_x=I_y=\frac{\pi \rho h}{12}(3(r_2^4-r_1^4)+h^2(r_2^2-r_1^2))$
Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass	File:Tetraaxial.gif	$I_{\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=\frac{1}{20}m{s}^2$ $I_{\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{1}{12}m{s}^2$
Regular octahedron of side s and mass m	File:Octahedral axis.gif	$I_{x,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{y,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{z,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{1}{6}m{s}^2$ $I_{x,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{y,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{z,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=\frac{1}{10}m{s}^2$
Regular dodecahedron of side s and mass m		$I_{x,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{y,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{z,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{39\varphi +28}{90}m{s}^2$ $I_{x,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{y,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{z,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=\frac{39\varphi +28}{150}m{s}^2$ (where $\varphi =\frac{1+\sqrt{5}}{2}$)
Regular icosahedron of side s and mass m		$I_{x,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{y,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=I_{z,\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}}=\frac{{\varphi }^2}{6}m{s}^2$ $I_{x,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{y,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=I_{z,\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{i}\mathrm{d}}=\frac{{\varphi }^2}{10}m{s}^2$
Hollow sphere of radius r and mass m.	File:Moment of inertia hollow sphere.svg	$I=\frac{2}{3}m{r}^2$
Solid sphere (ball) of radius r and mass m.	File:Moment of inertia solid sphere.svg	$I=\frac{2}{5}m{r}^2$
Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1.	File:Spherical shell moment of inertia.png	$I=\frac{2}{5}m\cdot \frac{r_2^5-r_1^5}{r_2^3-r_1^3}$	When the cavity radius r1 = 0, the object is a solid ball (above). When r1 becomes close to r2 the ratio $\frac{r_2^5-r_1^5}{r_2^3-r_1^3}$ approaches the value of $\frac{5}{3}{r}_2^2$, and in the limit the body becomes a thin hollow sphere with $I=\frac{2}{5}m\cdot \frac{5}{3}{r}_2^2=\frac{2}{3}m{r}_2^2.$
Right circular cone with radius r, height h and mass m	File:Moment of inertia cone.svg	$I_z=\frac{3}{10}m{r}^2$   About an axis passing through the tip: $I_x=I_y=m(\frac{3}{20}{r}^2+\frac{3}{5}{h}^2)$   About an axis passing through the base: $I_x=I_y=m(\frac{3}{20}{r}^2+\frac{1}{10}{h}^2)$ About an axis passing through the center of mass: $I_x=I_y=m(\frac{3}{20}{r}^2+\frac{3}{80}{h}^2)$ About a slanted axis passing through the apex (origin) and along the side generating line: $I_{\text{slant}}=\frac{3}{20}m{r}^2(1+5\frac{h^2}{h^2+r^2})$
Right circular hollow cone with radius r, height h and mass m	File:Moment of inertia cone.svg	$I_z=\frac{1}{2}m{r}^2$   $I_x=I_y=\frac{1}{4}m(r^2+2h^2)$
Torus with minor radius a, major radius b and mass m.	File:Torus cycles (labeled).png	About an axis passing through the center and perpendicular to the diameter: $\frac{1}{4}m(4b^2+3a^2)$   About a diameter: $\frac{1}{8}m(5a^2+4b^2)$
Ellipsoid (solid) of semiaxes a, b, and c with mass m	File:Ellipsoid tri-axial abc.svg	$I_x=\frac{1}{5}m(b^2+c^2)$ $I_y=\frac{1}{5}m(a^2+c^2)$ $I_z=\frac{1}{5}m(a^2+b^2)$
Thin rectangular plate of height h, width w and mass m (Axis of rotation at the end of the plate)	File:Recplaneoff.svg	$I_e=\frac{1}{12}m(4h^2+w^2)$
Thin rectangular plate of height h, width w and mass m (Axis of rotation at the center)	File:Recplane.svg	$I_c=\frac{1}{12}m(h^2+w^2)$
Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is rTemplate:Efn(Axis of rotation along a side of the plate)		$I=\frac{1}{3}m{r}^2$
Solid rectangular cuboid of height h, width w, and depth d, and mass m.	File:Moment of inertia solid rectangular prism.png	$I_h=\frac{1}{12}m(w^2+d^2)$ $I_w=\frac{1}{12}m(d^2+h^2)$ $I_d=\frac{1}{12}m(w^2+h^2)$	For a cube with sides $s$, $I=\frac{1}{6}m{s}^2$.
Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.	File:Moment of Inertia Cuboid.svg	$I=\frac{1}{6}m\left(\frac{W^2{D}^2+D^2{L}^2+W^2{L}^2}{W^2+D^2+L^2}\right)$	For a cube with sides $s$, $I=\frac{1}{6}m{s}^2$.
Tilted solid cuboid of depth d, width w, and length l, and mass m, rotating about the vertical axis (axis y as seen in figure).	File:Tiltedcuboid.pdf	$I=\frac{m}{12}(l^2{\cos }^2\beta +d^2{\sin }^2\beta +w^2)$	For a cube with sides $s$, $I=\frac{1}{6}m{s}^2$.
Triangle with vertices at the origin and at P and Q, with mass m, rotating about an axis perpendicular to the plane and passing through the origin.		$I=\frac{1}{6}m(\mathbf{𝐏}\cdot \mathbf{𝐏}+\mathbf{𝐏}\cdot \mathbf{𝐐}+\mathbf{𝐐}\cdot \mathbf{𝐐})$
Plane polygon with vertices P1, P2, P3, ..., PN and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through the origin.	File:Polygon Moment of Inertia.svg	$I=m\frac{\sum _{n=1}^N{Q}_n\Vert {\mathbf{𝐏}}_{n+1}\times {\mathbf{𝐏}}_n\Vert }{6\sum _{n=1}^N\Vert {\mathbf{𝐏}}_{n+1}\times {\mathbf{𝐏}}_n\Vert }$ where $Q_n={\Vert {\mathbf{𝐏}}_n\Vert }^2+{\mathbf{𝐏}}_n\cdot {\mathbf{𝐏}}_{n+1}+{\Vert {\mathbf{𝐏}}_{n+1}\Vert }^2$
Plane regular polygon with n-vertices and mass m uniformly distributed on its interior, rotating about an axis perpendicular to the plane and passing through its barycenter. R is the radius of the circumscribed circle.		$I=\frac{1}{2}m{R}^2(1-\frac{2}{3}{\sin }^2\left(\frac{\pi }{n}\right))$
An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane)	File:An isosceles triangle of mass M, vertex angle 2β and common-side length L.png	$I=\frac{1}{2}m{L}^2(1-\frac{2}{3}{\sin }^2\left(\beta \right))$
Infinite disk with mass distributed in a Bivariate Gaussian distribution on two axes around the axis of rotation with mass-density as a function of the position vector $\mathbf{𝐱}$ $$\rho (\mathbf{𝐱})=m\frac{\exp (-\frac{1}{2}{\mathbf{𝐱}}^{\mathrm{T}}{\mathrm{\Sigma }}^{-1}\mathbf{𝐱})}{\sqrt{(2\pi {)}^2|\mathrm{\Sigma }|}}$$	File:Gaussian 2D.png	$I=m\cdot \mathrm{tr}(\mathrm{\Sigma })$

List of 3D inertia tensors

This list of moment of inertia tensors is given for principal axes of each object.

To obtain the scalar moments of inertia I above, the tensor moment of inertia I is projected along some axis defined by a unit vector n according to the formula:

$$\mathbf{𝐧}\cdot \mathbf{𝐈}\cdot \mathbf{𝐧}\equiv n_i{I}_{ij}{n}_j,$$

where the dots indicate tensor contraction and the Einstein summation convention is used. In the above table, n would be the unit Cartesian basis e_x, e_y, e_z to obtain I_x, I_y, I_z respectively.

Description	Figure	Moment of inertia tensor
Solid sphere of radius r and mass m	File:Moment of inertia solid sphere.svg	$I=\left[\begin{array}{ccc}\frac{2}{5}m{r}^2& 0& 0\\ 0& \frac{2}{5}m{r}^2& 0\\ 0& 0& \frac{2}{5}m{r}^2\end{array}\right]$
Hollow sphere of radius r and mass m	File:Moment of inertia hollow sphere.svg	$I=\left[\begin{array}{ccc}\frac{2}{3}m{r}^2& 0& 0\\ 0& \frac{2}{3}m{r}^2& 0\\ 0& 0& \frac{2}{3}m{r}^2\end{array}\right]$
Solid ellipsoid of semi-axes a, b, c and mass m	File:Solid ellipsoid.svg	$I=\left[\begin{array}{ccc}\frac{1}{5}m(b^2+c^2)& 0& 0\\ 0& \frac{1}{5}m(a^2+c^2)& 0\\ 0& 0& \frac{1}{5}m(a^2+b^2)\end{array}\right]$
Right circular cone with radius r, height h and mass m, about the apex	File:Moment of inertia cone.svg	$I=\left[\begin{array}{ccc}\frac{3}{5}m{h}^2+\frac{3}{20}m{r}^2& 0& 0\\ 0& \frac{3}{5}m{h}^2+\frac{3}{20}m{r}^2& 0\\ 0& 0& \frac{3}{10}m{r}^2\end{array}\right]$
Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m	180x 180x	$I=\left[\begin{array}{ccc}\frac{1}{12}m(h^2+d^2)& 0& 0\\ 0& \frac{1}{12}m(w^2+d^2)& 0\\ 0& 0& \frac{1}{12}m(w^2+h^2)\end{array}\right]$
Slender rod along y-axis of length l and mass m about end	File:Moment of inertia rod end.svg	$I=\left[\begin{array}{ccc}\frac{1}{3}m{l}^2& 0& 0\\ 0& 0& 0\\ 0& 0& \frac{1}{3}m{l}^2\end{array}\right]$
Slender rod along y-axis of length l and mass m about center	File:Moment of inertia rod center.svg	$I=\left[\begin{array}{ccc}\frac{1}{12}m{l}^2& 0& 0\\ 0& 0& 0\\ 0& 0& \frac{1}{12}m{l}^2\end{array}\right]$
Solid cylinder of radius r, height h and mass m	File:Moment of inertia solid cylinder.svg	$I=\left[\begin{array}{ccc}\frac{1}{12}m(3r^2+h^2)& 0& 0\\ 0& \frac{1}{12}m(3r^2+h^2)& 0\\ 0& 0& \frac{1}{2}m{r}^2\end{array}\right]$
Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m	File:Moment of inertia thick cylinder h.svg	$I=\left[\begin{array}{ccc}\frac{1}{12}m(3(r_2^2+r_1^2)+h^2)& 0& 0\\ 0& \frac{1}{12}m(3(r_2^2+r_1^2)+h^2)& 0\\ 0& 0& \frac{1}{2}m(r_2^2+r_1^2)\end{array}\right]$

See also

• List of second moments of area
• Parallel axis theorem
• Perpendicular axis theorem

External links

• The inertia tensor of a tetrahedron
• Tutorial on deriving moment of inertia for common shapes