WARNING: the correctness of these equations are to be determined.
List of moments of inertia
| Description | Length / Area / Volume | Centroid / Center of mass |
Inertia tensor | Additional notes |
|---|---|---|---|---|
|
Mass change Multiplying the density of a body by $s=\frac{m_1}{m_0}$. |
Remain unchanged | Remain unchanged | $I_1=sI_0=I_0\dfrac{m_1}{m_0}$ | For the rest of this table, assume the mass of the body remains unchanged after transformation. |
|
Translation Translating a body by displacement $d$. |
Remain unchanged | $C_1=C_0+d$ |
$I_1=I+m\left(d^Td\ \mathbf{I}-dd^T\right)+m\left(2C_0^Td\ \mathbf{I}-C_0d^T-dC_0^T\right)$ $C_1=\mathbf{0}$: $I_1=I-m\left(d^Td\ \mathbf{I}-dd^T\right)$ |
Bold $\mathbf{I}$ represents the 3×3 identity matrix. |
|
Rotation about the origin Representing rotation using orthogonal matrix $\mathbf{R}$ where the columns of $\mathbf{R}$ are $\vec{i}, \vec{j}, \vec{k}$ after rotation. |
Remain unchanged | $C_1=\mathbf{R}C_0$ | $I_1=\mathbf{R}I_0\mathbf{R}^T$ | |
|
Scaling Scaling about the origin with factor $s>0$. |
$L_1=L_0s\\A_1=A_0s^2\\V_1=V_0s^3$ | $C_1=sC_0$ | $I_1=s^2I_0$ | If the density of the shape remains unchanged, then $I_1=s^3/s^4/s^5 \cdot I_0$. |
|
Stretching Scaling on unit axis $\mathbf{n}$ with a factor $s$. |
For solids only: $A_1=sA_0\\V_1=sV_0$ |
$C_1=C_0+(s-1)(C_0\cdot\mathbf{n})\mathbf{n}$ | ||
|
Reflection about xOy plane Reversing the signs of $z$ components of points in a body. |
Remain unchanged | Reflected about xOy plane | Remain unchanged | |
|
Reflection Reflection about a plane through the origin with unit normal $\mathbf{n}.$ |
Remain unchanged | $C_1=C_0-2(C_0\cdot\mathbf{n})\mathbf{n}$ | ||
|
Extrusion from 2D Extruding a shape on xOy plane on both positive and negative directions of z-axis by length $h$. (the result is a cylinder with height $2h$.) |
$S_1=2hL\\V_1=2hS$ | Remain unchanged | $I_1=I_0+\dfrac{1}{3}mh^2\begin{bmatrix}1&0&0\\0&1&0\\0&0&0\end{bmatrix}$ |
$I_0$ is a 3×3 matrix calculated in 3D. Reference rule 0. for mass change. |
| Description | Length / Area / Volume | Centroid / Center of mass |
Inertia tensor |
|---|---|---|---|
| Segment defined by endpoints $A$, $B$ | $L=\left|B-A\right|$ | $\dfrac{1}{2}\left(A+B\right)$ | $\dfrac{1}{3}m\left[\left(A^TA+A^TB+B^TB\right)\mathbf{I}-\left(AA^T+BB^T+\dfrac{1}{2}\left(AB^T+BA^T\right)\right)\right]$ |
| Triangle defined by the origin and vectors $\vec{a}$, $\vec{b}$ | $A=\dfrac{1}{2}\left|a\times b\right|$ | $\dfrac{1}{3}\left(a+b\right)$ | $\dfrac{1}{6}m\left[\left(a^Ta+a^Tb+b^Tb\right)\mathbf{I}-\left(aa^T+bb^T+\dfrac{1}{2}\left(ab^T+ba^T\right)\right)\right]$ |
| Triangle defined by vertices $A$, $B$, $C$ | $A=\dfrac{1}{2}\left|\left(B-A\right)\times\left(C-A\right)\right|$ | $\dfrac{1}{3}\left(A+B+C\right)$ | $\dfrac{1}{6}m\left[\left(A^TA+B^TB+C^TC+A^TB+A^TC+B^TC\right)\mathbf{I}\\ -\left(AA^T+BB^T+CC^T+\dfrac{1}{2}\left(AB^T+AC^T+BA^T+BC^T+CA^T+CB^T\right)\right)\right]$ |
| Parallelogram defined by point $p$ and vectors $a$, $b$ | $A=\left|a\times b\right|$ | $p+\dfrac{1}{2}a+\dfrac{1}{2}b$ | $m\left[\left(p^Tp+\dfrac{1}{3}\left(a^Ta+b^Tb\right)+a^Tp+b^Tp+\dfrac{1}{2}a^Tb\right)\mathbf{I}\\ -\left(pp^T+\dfrac{1}{3}\left(aa^T+bb^T\right)+\dfrac{1}{2}\left(ap^T+pa^T+bp^T+pb^T\right)+\dfrac{1}{4}\left(ab^T+ba^T\right)\right)\right]$ |
| Tetrahedron defined by the origin and vectors $a$, $b$, $c$ | $V=\dfrac{1}{6}\det\left[a,b,c\right]$ | $\dfrac{1}{4}\left(a+b+c\right)$ | $\dfrac{1}{10}m\left[\left(a^Ta+b^Tb+c^Tc+a^Tb+a^Tc+b^Tc\right)\mathbf{I}\\ -\left(aa^T+bb^T+cc^T+\dfrac{1}{2}\left(ab^T+ac^T+ba^T+bc^T+ca^T+cb^T\right)\right)\right]$ |
| Description | Length / Area / Volume | Moment of inertia / Inertia tensor | Graph | Additional notes |
|---|---|---|---|---|
| Rod | $L=2r$ | $m\begin{bmatrix}\frac{1}{3}r^2&0&0\\0&0&0\\0&0&\frac{1}{3}r^2\end{bmatrix}$ | $\dfrac{1}{3}r^2$ is equivalent to $\dfrac{1}{12}l^2$. | |
| Disk | $A=\pi r^2$ | $m\begin{bmatrix}\frac{1}{4}r^2&0&0\\0&\frac{1}{4}r^2&0\\0&0&\frac{1}{2}r^2\end{bmatrix}$ | ||
| Ring | $L=2\pi r$ | $m\begin{bmatrix}\frac{1}{2}r^2&0&0\\0&\frac{1}{2}r^2&0\\0&0&r^2\end{bmatrix}$ | ||
| Ring with width | $A=\pi(r_1^2-r_0^2)$ | $\dfrac{1}{4}m(r_1^2+r_0^2)\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}$ | ||
| Sector | $A=\alpha r^2$ | $\dfrac{1}{4}mr^2\begin{bmatrix}1-\frac{\sin(2\alpha)}{2\alpha}&0&0\\0&1+\frac{\sin(2\alpha)}{2\alpha}&0\\0&0&2\end{bmatrix}$ | $C=\left(\dfrac{2}{3}r\dfrac{\sin(\alpha)}{\alpha},0,0\right)$ | |
| Arc | $L=2\alpha r$ | $\dfrac{1}{2}mr^2\begin{bmatrix}1-\frac{\sin(2\alpha)}{2\alpha}&0&0\\0&1+\frac{\sin(2\alpha)}{2\alpha}&0\\0&0&2\end{bmatrix}$ | $C=\left(r\dfrac{\sin(\alpha)}{\alpha},0,0\right)$ | |
| Truncated circle | $A=\cos^{-1}\left(\dfrac{a}{r}\right)r^2-a\sqrt{r^2-a^2}$ | $C=\dfrac{1}{A}\left(\dfrac{2}{3}(r^2-a^2)^{\frac{3}{2}},0,0\right)$ | ||
| Cylinder | $V=2\pi r^2h$ | $\dfrac{1}{12}m\begin{bmatrix}3r^2+4h^2&0&0\\0&3r^2+4h^2&0\\0&0&6r^2\end{bmatrix}$ | The height of this cylinder is $2h$. For a cylinder with height $h$, change $4h^2$ in the inertia matrix to $h^2$. | |
| Sphere | $V=\dfrac{4}{3}\pi r^3$ | $m\begin{bmatrix}\frac{2}{5}r^2&0&0\\0&\frac{2}{5}r^2&0\\0&0&\frac{2}{5}r^2\end{bmatrix}$ | ||
| Cone | $V=\dfrac{1}{3}\pi r^2h$ | $\dfrac{3}{20}m\begin{bmatrix}r^2+4h^2&0&0\\0&r^2+4h^2&0\\0&0&2r^2\end{bmatrix}$ | $C=\left(0,0,\dfrac{3}{4}\right)$ | |
| Right-oriented cone | $V=\dfrac{1}{3}\pi r^2h$ | $\dfrac{1}{20}m\begin{bmatrix}3r^2+2h^2&0&0\\0&3r^2+2h^2&0\\0&0&6r^2\end{bmatrix}$ |
$C=\left(0,0,\dfrac{1}{4}h\right)$ Previous inertia subtracted by $m\begin{bmatrix}\frac{1}{2}h^2&0&0\\0&\frac{1}{2}h^2&0\\0&0&0\end{bmatrix}$ |
|
| Cone at center of mass | $V=\dfrac{1}{3}\pi r^2h$ | $\dfrac{3}{80}m\begin{bmatrix}4r^2+h^2&0&0\\0&4r^2+h^2&0\\0&0&8r^2\end{bmatrix}$ |
In the graph the cone has $z$ ranging from $-\frac{1}{4}h$ to $\frac{3}{4}h$. Reflection on xOy plane does not change its inertia. |
|
| Truncated cone | $V=\dfrac{1}{3}\pi r^2h_1(1-k^3)$ | $\dfrac{3}{20}m\dfrac{1-k^5}{1-k^3}\begin{bmatrix}r^2+4h_1^2&0&0\\0&r^2+4h_1^2&0\\0&0&2r^2\end{bmatrix}$ | $\frac{h_0}{h_1}=\frac{r_0}{r_1}=k$ |
$C=\left(0,0,\dfrac{3}{4}h_1\dfrac{1-k^4}{1-k^3}\right)$ $k=\dfrac{h_0}{h_1}=\dfrac{r_0}{r_1}\in[0,1)$ |
| Solid angle | $V=\dfrac{2}{3}\pi r^3(1-c)$ | $\dfrac{1}{10}mr^2\begin{bmatrix}c^2+c+4&0&0\\0&c^2+c+4&0\\0&0&-2(c^2+c-2)\end{bmatrix}$ | $c=\cos(\alpha)$ |
$C=\left(0,0,\dfrac{3}{8}r(1+c)\right)$ Semi-sphere is a special case with $c=0$. |
| Truncated sphere | ||||
| Torus | $V=2\pi^2Rr^2$ | $m\begin{bmatrix}\frac{1}{2}R^2+\frac{5}{8}r^2&0&0\\0&\frac{1}{2}R^2+\frac{5}{8}r^2&0\\0&0&R^2+\frac{3}{4}r^2\end{bmatrix}$ | ||
| Planar ellipse | $A=\pi ab$ | $\dfrac{1}{4}m\begin{bmatrix}b^2&0&0\\0&a^2&0\\0&0&a^2+b^2\end{bmatrix}$ | ||
| Planar elliptical ring | ||||
| Elliptical cylinder | $A=2\pi abh$ | $\dfrac{1}{12}m\begin{bmatrix}3b^2+4h^2&0&0\\0&3a^2+4h^2&0\\0&0&3a^2+3b^2\end{bmatrix}$ | Change $4h^2$ to $h^2$ for a cylinder with height $h$. | |
| Ellipsoid | $V=\dfrac{4}{3}\pi abc$ | $\dfrac{1}{5}m\begin{bmatrix}b^2+c^2&0&0\\0&a^2+c^2&0\\0&0&a^2+b^2\end{bmatrix}$ | ||
| Elliptical cone | $V=\dfrac{1}{3}\pi abh$ | $\dfrac{3}{20}m\begin{bmatrix}b^2+4h^2&0&0\\0&a^2+4h^2&0\\0&0&a^2+b^2\end{bmatrix}$ | $C=\left(0,0,\dfrac{3}{4}h\right)$ | |
| Truncated elliptical cone | $V=\dfrac{1}{3}\pi abh_1(1-k^3)$ | $\dfrac{3}{20}m\dfrac{1-k^5}{1-k^3}\begin{bmatrix}b^2+4h_1^2&0&0\\0&a^2+4h_1^2&0\\0&0&a^2+b^2\end{bmatrix}$ | $k=\frac{h_0}{h_1}$ | $C=\left(0,0,\dfrac{3}{4}h_1\dfrac{1-k^4}{1-k^3}\right)$ |
| Cylindrical shell | $A=4\pi rh$ | $\dfrac{1}{6}m\begin{bmatrix}3r^2+2h^2&0&0\\0&3r^2+2h^2&0\\0&0&6r^2\end{bmatrix}$ | Without bases | |
| Spherical shell | $A=4\pi r^2$ | $m\begin{bmatrix}\frac{2}{3}r^2&0&0\\0&\frac{2}{3}r^2&0\\0&0&\frac{2}{3}r^2\end{bmatrix}$ | ||
| Conical shell | $A=\pi r\sqrt{r^2+h^2}$ | $\dfrac{1}{4}m\begin{bmatrix}r^2+2h^2&0&0\\0&r^2+2h^2&0\\0&0&2r^2\end{bmatrix}$ |
Without base |
|
| Conical shell at center of mass | $A=\pi r\sqrt{r^2+h^2}$ | $\dfrac{1}{36}m\begin{bmatrix}9r^2+2h^2&0&0\\0&9r^2+2h^2&0\\0&0&18r^2\end{bmatrix}$ |
In the graph the cone has $z$ ranging from $-\frac{1}{3}h$ to $\frac{2}{3}h$. Reflection on xOy plane does not change its inertia. |
|
| Truncated spherical shell | $A=2\pi r^2(1-c)$ | $\dfrac{1}{6}mr^2\begin{bmatrix}c^2+c+4&0&0\\0&c^2+c+4&0\\0&0&-2(c^2+c-2)\end{bmatrix}$ | $c=\cos(\alpha)$ |
$C=\left(0,0,\dfrac{1}{2}r(1+c)\right)$ Semi-sphere is a special case with $c=0$. |
| Truncated conical shell | $A=\pi r_1\sqrt{r_1^2+h_1^2}\cdot(1-k^2)$ | $\dfrac{1}{4}m(1+k^2)\begin{bmatrix}r_1^2+2h_1^2&0&0\\0&r_1^2+2h_1^2&0\\0&0&2r_1^2\end{bmatrix}$ | $\frac{h_0}{h_1}=\frac{r_0}{r_1}=k$ |
Without bases |
| Elliptical cylindrical shell | ||||
| Ellipsoidal shell | ||||
| Elliptical conical shell | ||||
| Truncated elliptical conical shell | ||||
| Toric shell | $A=4\pi^2Rr$ | $m\begin{bmatrix}\frac{1}{2}R^2+\frac{5}{4}r^2&0&0\\0&\frac{1}{2}R^2+\frac{5}{4}r^2&0\\0&0&R^2+\frac{3}{2}r^2\end{bmatrix}$ | ||
| Cylindrical shell with thickness | $V=2\pi(r_1^2-r_0^2)h$ |
$\overline{r^2}=r_1^2+r_0^2$ $\dfrac{1}{12}m\begin{bmatrix}3\overline{r^2}+4h^2&0&0\\0&3\overline{r^2}+4h^2&0\\0&0&6\overline{r^2}\end{bmatrix}$ |
Change $4h^2$ in the inertia matrix to $h^2$ for a cylinder with height $h$. | |
| Spherical shell with thickness | $V=\dfrac{4}{3}\pi(r_1^3-r_0^3)$ | $\dfrac{2}{5}m\dfrac{r_1^5-r_0^5}{r_1^3-r_0^3}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ | ||
| Truncated conical shell with thickness | $V=\dfrac{\pi}{3}r_1^2h_1(1-k^2)(1-k^3)$ | $\dfrac{3}{20}m\dfrac{1-k^5}{1-k^3}\begin{bmatrix}r_1^2(1+k^2)+4h_1^2&0&0\\0&r_1^2(1+k^2)+4h_1^2&0\\0&0&2r_1^2(1+k^2)\end{bmatrix}$ | $C=(0,0,\dfrac{3}{4}h_1\dfrac{1-k^4}{1-k^3})$ | |
| Toric shell with thickness | $V=2\pi^2R(r_1^2-r_0^2)$ |
$\overline{r^2}=r_1^2+r_0^2$ $m\begin{bmatrix}\frac{1}{2}R^2+\frac{5}{8}\overline{r^2}&0&0\\0&\frac{1}{2}R^2+\frac{5}{8}\overline{r^2}&0\\0&0&R^2+\frac{3}{4}\overline{r^2}\end{bmatrix}$ |
$r_0$ and $r_1$ are inner and outer minor radiuses. |
| Description | Length / Area / Volume | Moment of inertia / Inertia tensor | Graph | Additional notes |
|---|---|---|---|---|
| Square | $A=4a^2$ | $m\begin{bmatrix}\frac{1}{3}a^2&0&0\\0&\frac{1}{3}a^2&0\\0&0&\frac{2}{3}a^2\end{bmatrix}$ | ||
| Rectangle | $A=4ab$ | $\dfrac{1}{3}m\begin{bmatrix}b^2&0&0\\0&a^2&0\\0&0&a^2+b^2\end{bmatrix}$ | ||
| Rectangle border | $L=4a+4b$ | $m\begin{bmatrix}\frac{b^2(a+\frac{1}{3}b)}{a+b}&0&0\\0&\frac{a^2(a+\frac{1}{3}b)}{a+b}&0\\0&0&\frac{1}{3}(a+b)^2\end{bmatrix}$ | ||
| Equilateral triangle | $A=\dfrac{3\sqrt{3}}{4}a^2$ | $\dfrac{5}{24}mr^2\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}$ | ||
| Regular polygon | $A=\dfrac{1}{2}r^2N\sin\left(\dfrac{2\pi}{N}\right)$ | $\dfrac{1}{12}mr^2\left(2+\cos\left(\dfrac{2\pi}{N}\right)\right)\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}$ | ||
| Regular polygon border | $L=N\sqrt{r^2+1-2r\cos\left(\dfrac{2\pi}{N}\right)}$ | $\dfrac{1}{6}mr^2\left(2+\cos\left(\dfrac{2\pi}{N}\right)\right)\begin{bmatrix}1&0&0\\0&1&0\\0&0&2\end{bmatrix}$ | ||
| Cube | $V=8a^3$ | $m\begin{bmatrix}\frac{2}{3}a^2&0&0\\0&\frac{2}{3}a^2&0\\0&0&\frac{2}{3}a^2\end{bmatrix}$ | ||
| Box (cuboid) | $V=8abc$ | $\dfrac{1}{3}m\begin{bmatrix}b^2+c^2&0&0\\0&a^2+c^2&0\\0&0&a^2+b^2\end{bmatrix}$ | ||
| Cube shell | $A=24a^2$ | $\dfrac{10}{9}ma^2\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ | ||
| Box shell | $A=8(ab+ac+bc)$ | $\dfrac{5}{9}m\begin{bmatrix}b^2+c^2&0&0\\0&a^2+c^2&0\\0&0&a^2+b^2\end{bmatrix}$ | ||
| Cube shell with thickness | $V=8(a_1^3-a_0^3)$ | $\dfrac{2}{3}m\dfrac{a_1^5-a_0^5}{a_1^3-a_0^3}\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$ |
Additional notes
Most integrations are done manually and checked numerically.
Formulas are rendered with MathJax. A script has been used to generate latex. Graphs are generated with Javascript.
Please check my math and English.