List of moment of inertia tensors

List of 3D inertia tensors

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

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

See also

• Moment of inertia
• List of area moments of inertia
• The inertia tensor of a triangle in three-dimensional space
• The inertia tensor of a tetrahedron