List of moments of inertia

Point mass M at a distance r from the axis of rotation.

I = M r 2 {\displaystyle I=Mr^{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.

I = m 1 m 2 m 1 + m 2 x 2 = μ x 2 {\displaystyle 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.

I c e n t e r = 1 12 m L 2 {\displaystyle I_{\mathrm {center} }={\frac {1}{12}}mL^{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.

I e n d = 1 3 m L 2 {\displaystyle I_{\mathrm {end} }={\frac {1}{3}}mL^{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.

I z = m r 2 {\displaystyle I_{z}=mr^{2}\!} I x = I y = 1 2 m r 2 {\displaystyle I_{x}=I_{y}={\frac {1}{2}}mr^{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.

I z = 1 2 m r 2 {\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!} I x = I y = 1 4 m r 2 {\displaystyle I_{x}=I_{y}={\frac {1}{4}}mr^{2}\,\!}

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.

A uniform annulus (disk with a concentric hole) of mass m, inner radius r1 and outer radius r2

I z = 1 2 m ( r 1 2 + r 2 2 ) {\displaystyle I_{z}={\frac {1}{2}}m(r_{1}^{2}+r_{2}^{2})} I x = I y = 1 4 m ( r 1 2 + r 2 2 ) {\displaystyle I_{x}=I_{y}={\frac {1}{4}}m(r_{1}^{2}+r_{2}^{2})}

An annulus with a constant area density ρ A {\displaystyle \rho _{A}}

I z = 1 2 π ρ A ( r 2 4 − r 1 4 ) {\displaystyle I_{z}={\frac {1}{2}}\pi \rho _{A}(r_{2}^{4}-r_{1}^{4})} I x = I y = 1 4 π ρ A ( r 2 4 − r 1 4 ) {\displaystyle 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.

I = m r 2 {\displaystyle I=mr^{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.

I z = 1 2 m r 2 {\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!}    I x = I y = 1 12 m ( 3 r 2 + h 2 ) {\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3r^{2}+h^{2}\right)}

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.

I z = 1 2 m ( r 2 2 + r 1 2 ) = m r 2 2 ( 1 − t + t 2 2 ) {\displaystyle I_{z}={\frac {1}{2}}m\left(r_{2}^{2}+r_{1}^{2}\right)=mr_{2}^{2}\left(1-t+{\frac {t^{2}}{2}}\right)}    where t = r 2 − r 1 r 2 {\displaystyle t={\frac {r_{2}-r_{1}}{r_{2}}}} is a normalized thickness ratio; I x = I y = 1 12 m ( 3 ( r 2 2 + r 1 2 ) + h 2 ) {\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3\left(r_{2}^{2}+r_{1}^{2}\right)+h^{2}\right)} [citation needed]

The given formula is for the x y {\displaystyle 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 = h 2 , {\displaystyle d={\frac {h}{2}},} then by the parallel axis theorem the following formula applies: I x = I y = 1 12 m ( 3 ( r 2 2 + r 1 2 ) + 4 h 2 ) {\displaystyle I_{x}=I_{y}={\frac {1}{12}}m\left(3\left(r_{2}^{2}+r_{1}^{2}\right)+4h^{2}\right)}

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)}

Regular tetrahedron of side s and mass m with an axis of rotation passing through a tetrahedron's vertex and its center of mass

I s o l i d = 1 20 m s 2 {\displaystyle I_{\mathrm {solid} }={\frac {1}{20}}ms^{2}\,\!} I h o l l o w = 1 12 m s 2 {\displaystyle I_{\mathrm {hollow} }={\frac {1}{12}}ms^{2}\,\!}

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 = 1 6 m s 2 {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {1}{6}}ms^{2}\,\!} I x , s o l i d = I y , s o l i d = I z , s o l i d = 1 10 m s 2 {\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {1}{10}}ms^{2}\,\!}

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 = 39 ϕ + 28 90 m s 2 {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {39\phi +28}{90}}ms^{2}} I x , s o l i d = I y , s o l i d = I z , s o l i d = 39 ϕ + 28 150 m s 2 {\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {39\phi +28}{150}}ms^{2}\,\!} (where ϕ = 1 + 5 2 {\displaystyle \phi ={\frac {1+{\sqrt {5}}}{2}}} )

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 = ϕ 2 6 m s 2 {\displaystyle I_{x,\mathrm {hollow} }=I_{y,\mathrm {hollow} }=I_{z,\mathrm {hollow} }={\frac {\phi ^{2}}{6}}ms^{2}} I x , s o l i d = I y , s o l i d = I z , s o l i d = ϕ 2 10 m s 2 {\displaystyle I_{x,\mathrm {solid} }=I_{y,\mathrm {solid} }=I_{z,\mathrm {solid} }={\frac {\phi ^{2}}{10}}ms^{2}\,\!}

Hollow sphere of radius r and mass m.

I = 2 3 m r 2 {\displaystyle I={\frac {2}{3}}mr^{2}\,\!}

Solid sphere (ball) of radius r and mass m.

I = 2 5 m r 2 {\displaystyle I={\frac {2}{5}}mr^{2}\,\!}

Sphere (shell) of radius r2 and mass m, with centered spherical cavity of radius r1.

I = 2 5 m ⋅ r 2 5 − r 1 5 r 2 3 − r 1 3 {\displaystyle 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 r 2 5 − r 1 5 r 2 3 − r 1 3 {\displaystyle {\frac {r_{2}^{5}-r_{1}^{5}}{r_{2}^{3}-r_{1}^{3}}}} approaches the value of 5 3 r 2 2 {\displaystyle {\frac {5}{3}}r_{2}^{2}} , and in the limit the body becomes a thin hollow sphere with I = 2 5 m ⋅ 5 3 r 2 2 = 2 3 m r 2 2 . {\displaystyle I={\frac {2}{5}}m\cdot {\frac {5}{3}}r_{2}^{2}={\frac {2}{3}}mr_{2}^{2}.}

Right circular cone with radius r, height h and mass m

I z = 3 10 m r 2 {\displaystyle I_{z}={\frac {3}{10}}mr^{2}\,\!}    About an axis passing through the tip: I x = I y = m ( 3 20 r 2 + 3 5 h 2 ) {\displaystyle I_{x}=I_{y}=m\left({\frac {3}{20}}r^{2}+{\frac {3}{5}}h^{2}\right)\,\!}    About an axis passing through the base: I x = I y = m ( 3 20 r 2 + 1 10 h 2 ) {\displaystyle I_{x}=I_{y}=m\left({\frac {3}{20}}r^{2}+{\frac {1}{10}}h^{2}\right)\,\!} About an axis passing through the center of mass: I x = I y = m ( 3 20 r 2 + 3 80 h 2 ) {\displaystyle I_{x}=I_{y}=m\left({\frac {3}{20}}r^{2}+{\frac {3}{80}}h^{2}\right)\,\!} About a slanted axis passing through the apex (origin) and along the side generating line: I slant = 3 20 m r 2 ( 1 + 5 h 2 h 2 + r 2 ) {\displaystyle I_{\text{slant}}={\frac {3}{20}}mr^{2}\left(1+5{\frac {h^{2}}{h^{2}+r^{2}}}\right)\,\!}

Right circular hollow cone with radius r, height h and mass m

I z = 1 2 m r 2 {\displaystyle I_{z}={\frac {1}{2}}mr^{2}\,\!}    I x = I y = 1 4 m ( r 2 + 2 h 2 ) {\displaystyle I_{x}=I_{y}={\frac {1}{4}}m\left(r^{2}+2h^{2}\right)\,\!}

Torus with minor radius a, major radius b and mass m.

About an axis passing through the center and perpendicular to the diameter: 1 4 m ( 4 b 2 + 3 a 2 ) {\displaystyle {\frac {1}{4}}m\left(4b^{2}+3a^{2}\right)}   About a diameter: 1 8 m ( 5 a 2 + 4 b 2 ) {\displaystyle {\frac {1}{8}}m\left(5a^{2}+4b^{2}\right)}

Ellipsoid (solid) of semiaxes a, b, and c with mass m

I x = 1 5 m ( b 2 + c 2 ) {\displaystyle I_{x}={\frac {1}{5}}m\left(b^{2}+c^{2}\right)\,\!} I y = 1 5 m ( a 2 + c 2 ) {\displaystyle I_{y}={\frac {1}{5}}m\left(a^{2}+c^{2}\right)\,\!} I z = 1 5 m ( a 2 + b 2 ) {\displaystyle I_{z}={\frac {1}{5}}m\left(a^{2}+b^{2}\right)\,\!}

Thin rectangular plate of height h, width w and mass m(Axis of rotation at the end of the plate)

I e = 1 12 m ( 4 h 2 + w 2 ) {\displaystyle I_{e}={\frac {1}{12}}m\left(4h^{2}+w^{2}\right)\,\!}

Thin rectangular plate of height h, width w and mass m(Axis of rotation at the center)

I c = 1 12 m ( h 2 + w 2 ) {\displaystyle I_{c}={\frac {1}{12}}m\left(h^{2}+w^{2}\right)\,\!}

Thin rectangular plate of mass m, length of side adjacent to side containing axis of rotation is r(Axis of rotation along a side of the plate)

I = 1 3 m r 2 {\displaystyle I={\frac {1}{3}}mr^{2}}

Solid rectangular cuboid of height h, width w, and depth d, and mass m.

I h = 1 12 m ( w 2 + d 2 ) {\displaystyle I_{h}={\frac {1}{12}}m\left(w^{2}+d^{2}\right)} I w = 1 12 m ( d 2 + h 2 ) {\displaystyle I_{w}={\frac {1}{12}}m\left(d^{2}+h^{2}\right)} I d = 1 12 m ( w 2 + h 2 ) {\displaystyle I_{d}={\frac {1}{12}}m\left(w^{2}+h^{2}\right)}

For a cube with sides s {\displaystyle s} , I = 1 6 m s 2 {\displaystyle I={\frac {1}{6}}ms^{2}\,\!} .

Solid cuboid of height D, width W, and length L, and mass m, rotating about the longest diagonal.

I = 1 6 m ( W 2 D 2 + D 2 L 2 + W 2 L 2 W 2 + D 2 + L 2 ) {\displaystyle 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 {\displaystyle s} , I = 1 6 m s 2 {\displaystyle I={\frac {1}{6}}ms^{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).

I = m 12 ( l 2 cos 2 ⁡ β + d 2 sin 2 ⁡ β + w 2 ) {\displaystyle I={\frac {m}{12}}\left(l^{2}\cos ^{2}\beta +d^{2}\sin ^{2}\beta +w^{2}\right)}

For a cube with sides s {\displaystyle s} , I = 1 6 m s 2 {\displaystyle I={\frac {1}{6}}ms^{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 = 1 6 m ( P ⋅ P + P ⋅ Q + Q ⋅ Q ) {\displaystyle 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.

I = m ∑ n = 1 N Q n ‖ P n + 1 × P n ‖ 6 ∑ n = 1 N ‖ P n + 1 × P n ‖ {\displaystyle I=m{\frac {\sum \limits _{n=1}^{N}Q_{n}\left\|\mathbf {P} _{n+1}\times \mathbf {P} _{n}\right\|}{6\sum \limits _{n=1}^{N}\left\|\mathbf {P} _{n+1}\times \mathbf {P} _{n}\right\|}}} where Q n = ‖ P n ‖ 2 + P n ⋅ P n + 1 + ‖ P n + 1 ‖ 2 {\displaystyle Q_{n}=\left\|\mathbf {P} _{n}\right\|^{2}+\mathbf {P} _{n}\cdot \mathbf {P} _{n+1}+\left\|\mathbf {P} _{n+1}\right\|^{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 = 1 2 m R 2 ( 1 − 2 3 sin 2 ⁡ ( π n ) ) {\displaystyle I={\frac {1}{2}}mR^{2}\left(1-{\frac {2}{3}}\sin ^{2}\left({\tfrac {\pi }{n}}\right)\right)}

An isosceles triangle of mass M, vertex angle 2β and common-side length L (axis through tip, perpendicular to plane)

I = 1 2 m L 2 ( 1 − 2 3 sin 2 ⁡ ( β ) ) {\displaystyle I={\frac {1}{2}}mL^{2}\left(1-{\frac {2}{3}}\sin ^{2}\left(\beta \right)\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 x {\displaystyle {\mathbf {x} }} ρ ( x ) = m exp ⁡ ( − 1 2 x T Σ − 1 x ) ( 2 π ) 2 | Σ | {\displaystyle \rho ({\mathbf {x} })=m{\frac {\exp \left(-{\frac {1}{2}}{\mathbf {x} }^{\mathrm {T} }{\boldsymbol {\Sigma }}^{-1}{\mathbf {x} }\right)}{\sqrt {(2\pi )^{2}|{\boldsymbol {\Sigma }}|}}}}

I = m ⋅ tr ⁡ ( Σ ) {\displaystyle I=m\cdot \operatorname {tr} ({\boldsymbol {\Sigma }})\,\!}

Solid sphere of radius r and mass m

I = [ 2 5 m r 2 0 0 0 2 5 m r 2 0 0 0 2 5 m r 2 ] {\displaystyle I={\begin{bmatrix}{\frac {2}{5}}mr^{2}&0&0\\0&{\frac {2}{5}}mr^{2}&0\\0&0&{\frac {2}{5}}mr^{2}\end{bmatrix}}}

Hollow sphere of radius r and mass m

I = [ 2 3 m r 2 0 0 0 2 3 m r 2 0 0 0 2 3 m r 2 ] {\displaystyle I={\begin{bmatrix}{\frac {2}{3}}mr^{2}&0&0\\0&{\frac {2}{3}}mr^{2}&0\\0&0&{\frac {2}{3}}mr^{2}\end{bmatrix}}}

Solid ellipsoid of semi-axes a, b, c and mass m

I = [ 1 5 m ( b 2 + c 2 ) 0 0 0 1 5 m ( a 2 + c 2 ) 0 0 0 1 5 m ( a 2 + b 2 ) ] {\displaystyle I={\begin{bmatrix}{\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{bmatrix}}}

Right circular cone with radius r, height h and mass m, about the apex

I = [ 3 5 m h 2 + 3 20 m r 2 0 0 0 3 5 m h 2 + 3 20 m r 2 0 0 0 3 10 m r 2 ] {\displaystyle I={\begin{bmatrix}{\frac {3}{5}}mh^{2}+{\frac {3}{20}}mr^{2}&0&0\\0&{\frac {3}{5}}mh^{2}+{\frac {3}{20}}mr^{2}&0\\0&0&{\frac {3}{10}}mr^{2}\end{bmatrix}}}

Solid cuboid of width w (x-direction), height h (y-direction), depth d (z-direction), and mass m

180x

I = [ 1 12 m ( h 2 + d 2 ) 0 0 0 1 12 m ( w 2 + d 2 ) 0 0 0 1 12 m ( w 2 + h 2 ) ] {\displaystyle I={\begin{bmatrix}{\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{bmatrix}}}

Slender rod along y-axis of length l and mass m about end

I = [ 1 3 m l 2 0 0 0 0 0 0 0 1 3 m l 2 ] {\displaystyle I={\begin{bmatrix}{\frac {1}{3}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{3}}ml^{2}\end{bmatrix}}}

Slender rod along y-axis of length l and mass m about center

I = [ 1 12 m l 2 0 0 0 0 0 0 0 1 12 m l 2 ] {\displaystyle I={\begin{bmatrix}{\frac {1}{12}}ml^{2}&0&0\\0&0&0\\0&0&{\frac {1}{12}}ml^{2}\end{bmatrix}}}

Solid cylinder of radius r, height h and mass m

I = [ 1 12 m ( 3 r 2 + h 2 ) 0 0 0 1 12 m ( 3 r 2 + h 2 ) 0 0 0 1 2 m r 2 ] {\displaystyle I={\begin{bmatrix}{\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}}mr^{2}\end{bmatrix}}}

Thick-walled cylindrical tube with open ends, of inner radius r1, outer radius r2, length h and mass m

I = [ 1 12 m ( 3 ( r 2 2 + r 1 2 ) + h 2 ) 0 0 0 1 12 m ( 3 ( r 2 2 + r 1 2 ) + h 2 ) 0 0 0 1 2 m ( r 2 2 + r 1 2 ) ] {\displaystyle I={\begin{bmatrix}{\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{bmatrix}}}