List of moments of inertia

Table columns:
Description
Figure
Moment(s) of inertia

Table rows:
cells: Point mass m at a distance r from the axis of rotation. 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.
I  =  m  r  2  {\displaystyle I=mr^{2}}
cells: Two point masses, M and m, 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.
I  =  M  m  M  +  m  x  2  =  μ  x  2  {\displaystyle I={\frac {Mm}{M\!+\!m}}x^{2}=\mu x^{2}}
cells: Rod of length L and mass m, rotating about its center. 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.
I  c  e  n  t  e  r  =  m  L  2  12  {\displaystyle I_{\mathrm {center} }={\frac {mL^{2}}{12}}\,\!}  [1]
cells: Rod of length L and mass m, rotating about one end. 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.
I  e  n  d  =  m  L  2  3  {\displaystyle I_{\mathrm {end} }={\frac {mL^{2}}{3}}\,\!}  [1]
cells: Thin circular hoop of radius r and mass m. 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.
I  z  =  m  r  2  {\displaystyle I_{z}=mr^{2}\!}  I  x  =  I  y  =  m  r  2  2  {\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{2}}\,\!}
cells: Thin, solid disk of radius r and mass m. This is a special case of the solid cylinder, with h = 0. That I  x  =  I  y  =  I  z  2  {\displaystyle I_{x}=I_{y}={\frac {I_{z}}{2}}\,}  is a consequence of the perpendicular axis theorem.
I  z  =  m  r  2  2  {\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}  I  x  =  I  y  =  m  r  2  4  {\displaystyle I_{x}=I_{y}={\frac {mr^{2}}{4}}\,\!}
cells: Thin cylindrical shell with open ends, of radius r and mass m. This expression assumes that the shell thickness is negligible. It is a special case of the thick-walled cylindrical tube for r1 = r2. Also, a point mass m at the end of a rod of length r has this same moment of inertia and the value r is called the radius of gyration.
I  ≈  m  r  2  {\displaystyle I\approx mr^{2}\,\!}  [1]
cells: Solid cylinder of radius r, height h and mass m. This is a special case of the thick-walled cylindrical tube, with r1 = 0.
I  z  =  m  r  2  2  {\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}  [1]  I  x  =  I  y  =  m  12  (  3  r  2  +  h  2  )  {\displaystyle I_{x}=I_{y}={\frac {m}{12}}\left(3r^{2}+h^{2}\right)}
cells: Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m.
I  z  =  m  2  (  r  1  2  +  r  2  2  )  =  m  r  2  2  (  1  −  t  +  t  2  2  )  {\displaystyle I_{z}={\frac {m}{2}}\left(r_{1}^{2}+r_{2}^{2}\right)=mr_{2}^{2}\left(1-t+{\frac {t^{2}}{2}}\right)}  [1] [2] where t = (r2–r1)/r2 is a normalized thickness ratio;  I  x  =  I  y  =  m  12  (  3  (  r  2  2  +  r  1  2  )  +  h  2  )  {\displaystyle I_{x}=I_{y}={\frac {m}{12}}\left(3\left({r_{2}}^{2}+{r_{1}}^{2}\right)+h^{2}\right)}
cells: With a density of ρ and the same geometry
I  z  =  π  ρ  h  2  (  r  2  4  −  r  1  4  )  {\displaystyle I_{z}={\frac {\pi \rho h}{2}}\left({r_{2}}^{4}-{r_{1}}^{4}\right)}  I  x  =  I  y  =  π  ρ  h  12  (  3  (  r  2  4  −  r  1  4  )  +  h  2  (  r  2  2  −  r  1  2  )  )  {\displaystyle I_{x}=I_{y}={\frac {\pi \rho h}{12}}\left(3({r_{2}}^{4}-{r_{1}}^{4})+h^{2}({r_{2}}^{2}-{r_{1}}^{2})\right)}
cells: Regular tetrahedron of side s and mass m
I  s  o  l  i  d  =  m  s  2  20  {\displaystyle I_{\mathrm {solid} }={\frac {ms^{2}}{20}}\,\!}  I  h  o  l  l  o  w  =  m  s  2  12  {\displaystyle I_{\mathrm {hollow} }={\frac {ms^{2}}{12}}\,\!}  [3]
cells: Regular octahedron of side s and mass m
I  x  ,  h  o  l  l  o  w  =  I  y  ,  h  o  l  l  o  w  =  I  z  ,  h  o  l  l  o  w  =  m  s  2  6  {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {ms^{2}}{6}}\,\!}  [3]  I  x  ,  s  o  l  i  d  =  I  y  ,  s  o  l  i  d  =  I  z  ,  s  o  l  i  d  =  m  s  2  10  {\displaystyle I_{x,solid}=I_{y,solid}=I_{z,solid}={\frac {ms^{2}}{10}}\,\!}  [3]
cells: Regular dodecahedron of side s and mass m
I  x  ,  h  o  l  l  o  w  =  I  y  ,  h  o  l  l  o  w  =  I  z  ,  h  o  l  l  o  w  =  m  s  2  (  39  ϕ  +  28  )  90  {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {ms^{2}(39\phi +28)}{90}}}  I  x  ,  s  o  l  i  d  =  I  y  ,  s  o  l  i  d  =  I  z  ,  s  o  l  i  d  =  m  s  2  (  39  ϕ  +  28  )  150  {\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {ms^{2}(39\phi +28)}{150}}\,\!}  (where ϕ  =  1  +  5  2  {\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}}  ) [3]
cells: Regular icosahedron of side s and mass m
I  x  ,  h  o  l  l  o  w  =  I  y  ,  h  o  l  l  o  w  =  I  z  ,  h  o  l  l  o  w  =  m  s  2  ϕ  2  6  {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {ms^{2}\phi ^{2}}{6}}}  I  x  ,  s  o  l  i  d  =  I  y  ,  s  o  l  i  d  =  I  z  ,  s  o  l  i  d  =  m  s  2  ϕ  2  10  {\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {ms^{2}\phi ^{2}}{10}}\,\!}  [3]
cells: Hollow sphere of radius r and mass m. A hollow sphere can be taken to be made up of two stacks of infinitesimally thin, circular hoops, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r).
I  =  2  m  r  2  3  {\displaystyle I={\frac {2mr^{2}}{3}}\,\!}  [1]
cells: Solid sphere (ball) of radius r and mass m. A sphere can be taken to be made up of two stacks of infinitesimally thin, solid discs, where the radius differs from 0 to r (or a single stack, where the radius differs from -r to r).
I  =  2  m  r  2  5  {\displaystyle I={\frac {2mr^{2}}{5}}\,\!}  [1]
cells: Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1. When the cavity radius r1 = 0, the object is a solid ball (above). When r1 = r2, (  r  2  5  −  r  1  5  r  2  3  −  r  1  3  )  =  5  3  r  2  2  {\displaystyle \left({\frac {{r_{2}}^{5}-{r_{1}}^{5}}{{r_{2}}^{3}-{r_{1}}^{3}}}\right)={\frac {5}{3}}{r_{2}}^{2}}  , and the object is a hollow sphere.
I  =  2  m  5  (  r  2  5  −  r  1  5  r  2  3  −  r  1  3  )  {\displaystyle I={\frac {2m}{5}}\left({\frac {{r_{2}}^{5}-{r_{1}}^{5}}{{r_{2}}^{3}-{r_{1}}^{3}}}\right)\,\!}  [1]
cells: Right circular cone with radius r, height h and mass m
I  z  =  3  m  r  2  10  {\displaystyle I_{z}={\frac {3mr^{2}}{10}}\,\!}  [4]  I  x  =  I  y  =  3  m  20  (  r  2  +  4  h  2  )  {\displaystyle I_{x}=I_{y}={\frac {3m}{20}}\left(r^{2}+4h^{2}\right)\,\!}  [4]
cells: Right circular hollow cone with radius r, height h and mass m
I  z  =  m  r  2  2  {\displaystyle I_{z}={\frac {mr^{2}}{2}}\,\!}  [4]  I  x  =  I  y  =  m  4  (  r  2  +  2  h  2  )  {\displaystyle I_{x}=I_{y}={\frac {m}{4}}\left(r^{2}+2h^{2}\right)\,\!}  [4]
cells: Torus with minor radius a, major radius b and mass m.
About an axis passing through the center and perpendicular to the diameter: m  4  (  4  b  2  +  3  a  2  )  {\displaystyle {\frac {m}{4}}\left(4b^{2}+3a^{2}\right)}  [5] About a diameter: m  8  (  5  a  2  +  4  b  2  )  {\displaystyle {\frac {m}{8}}\left(5a^{2}+4b^{2}\right)}  [5]
cells: Ellipsoid (solid) of semiaxes a, b, and c with mass m
I  a  =  m  5  (  b  2  +  c  2  )  {\displaystyle I_{a}={\frac {m}{5}}\left(b^{2}+c^{2}\right)\,\!}  I  b  =  m  5  (  a  2  +  c  2  )  {\displaystyle I_{b}={\frac {m}{5}}\left(a^{2}+c^{2}\right)\,\!}  I  c  =  m  5  (  a  2  +  b  2  )  {\displaystyle I_{c}={\frac {m}{5}}\left(a^{2}+b^{2}\right)\,\!}
cells: Thin rectangular plate of height h, width w and mass m (Axis of rotation at the end of the plate)
I  e  =  m  12  (  4  h  2  +  w  2  )  {\displaystyle I_{e}={\frac {m}{12}}\left(4h^{2}+w^{2}\right)\,\!}
cells: Thin rectangular plate of height h, width w and mass m (Axis of rotation at the center)
I  c  =  m  12  (  h  2  +  w  2  )  {\displaystyle I_{c}={\frac {m}{12}}\left(h^{2}+w^{2}\right)\,\!}  [1]
cells: Solid cuboid of height h, width w, and depth d, and mass m. For a similarly oriented cube with sides of length s  {\displaystyle s}  , I  C  M  =  m  s  2  6  {\displaystyle I_{CM}={\frac {ms^{2}}{6}}\,\!}
I  h  =  m  12  (  w  2  +  d  2  )  {\displaystyle I_{h}={\frac {m}{12}}\left(w^{2}+d^{2}\right)}  I  w  =  m  12  (  d  2  +  h  2  )  {\displaystyle I_{w}={\frac {m}{12}}\left(d^{2}+h^{2}\right)}  I  d  =  m  12  (  w  2  +  h  2  )  {\displaystyle I_{d}={\frac {m}{12}}\left(w^{2}+h^{2}\right)}
cells: Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal. For a cube with sides s  {\displaystyle s}  , I  =  m  s  2  6  {\displaystyle I={\frac {ms^{2}}{6}}\,\!}  .
I  =  m  6  (  W  2  D  2  +  D  2  L  2  +  W  2  L  2  W  2  +  D  2  +  L  2  )  {\displaystyle I={\frac {m}{6}}\left({\frac {W^{2}D^{2}+D^{2}L^{2}+W^{2}L^{2}}{W^{2}+D^{2}+L^{2}}}\right)}
cells: 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  =  m  6  (  P  ⋅  P  +  P  ⋅  Q  +  Q  ⋅  Q  )  {\displaystyle I={\frac {m}{6}}(\mathbf {P} \cdot \mathbf {P} +\mathbf {P} \cdot \mathbf {Q} +\mathbf {Q} \cdot \mathbf {Q} )}
cells: 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.
I  =  m  6  (  ∑  n  =  1  N  ∥  P  n  +  1  ×  P  n  ∥  (  (  P  n  ⋅  P  n  )  +  (  P  n  ⋅  P  n  +  1  )  +  (  P  n  +  1  ⋅  P  n  +  1  )  )  ∑  n  =  1  N  ∥  P  n  +  1  ×  P  n  ∥  )  {\displaystyle I={\frac {m}{6}}\left({\frac {\sum \limits _{n=1}^{N}\|\mathbf {P} _{n+1}\times \mathbf {P} _{n}\|\left(\left(\mathbf {P} _{n}\cdot \mathbf {P} _{n}\right)+\left(\mathbf {P} _{n}\cdot \mathbf {P} _{n+1}\right)+\left(\mathbf {P} _{n+1}\cdot \mathbf {P} _{n+1}\right)\right)}{\sum \limits _{n=1}^{N}\|\mathbf {P} _{n+1}\times \mathbf {P} _{n}\|}}\right)}
cells: 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 the origin. a stands for side length.
I  =  m  a  2  24  (  1  +  3  cot  2  ⁡  (  π  n  )  )  {\displaystyle I={\frac {ma^{2}}{24}}\left(1+3\cot ^{2}\left({\tfrac {\pi }{n}}\right)\right)}  [6]
cells: Infinite disk with mass normally distributed on two axes around the axis of rotation with mass-density as a function of x and y:  ρ  (  x  ,  y  )  =  m  2  π  a  b  e  −  (  (  x  /  a  )  2  +  (  y  /  b  )  2  )  /  2  ,  {\displaystyle \rho (x,y)={\tfrac {m}{2\pi ab}}\,e^{-((x/a)^{2}+(y/b)^{2})/2}\,,}
I  =  m  (  a  2  +  b  2  )  {\displaystyle I=m(a^{2}+b^{2})\,\!}
cells: Uniform disk about an axis perpendicular to its edge.
I  =  3  m  r  2  2  {\displaystyle I={\frac {3mr^{2}}{2}}}  [7]

Questions about this table:
originalText: moment of inertia of a rod about a point
answer: answerTexts: radius of gyration