Free Mass Moment of Inertia Calculator

Mass Moment of Inertia Calculator — Your Versatile Rotational Inertia Tool

The mass moment of inertia calculator described here works as an online rotational inertia calculator that quickly determines the angular mass (rotational inertia) of a wide range of homogeneous objects. By selecting a shape and supplying its dimensions and mass, you obtain the moment of inertia in units such as $\text{kg·m}^{2}$ or $\text{lb·ft·s}^{2}$. This physics inertia calculator covers 23 standard geometries and is suitable for engineering design, homework problems, and quick design checks.

What Is Rotational Inertia?

Rotational inertia — also called the mass moment of inertia or angular mass — measures how difficult it is to alter the rotational speed of a rigid body about a given axis. It plays the same role in rotational dynamics that ordinary mass does in linear motion: a larger inertia demands a larger torque to achieve the same angular acceleration. The axis location and orientation are critical because the same object can yield very different inertia values for different axes.

How to Compute Moment of Inertia

For a point particle, the relation is:

$$I = m r^{2},$$

where $m$ is the mass and $r$ the perpendicular distance to the rotation axis. For a system of discrete particles, the total inertia is the sum:

$$I = \sum_{i} m_{i} r_{i}^{2}.$$

When mass is distributed continuously, the formula becomes a volume integral:

$$I = \int_{V} \rho \, r_{\perp}^{2} \, dV.$$

Instead of evaluating integrals by hand, you can pick a pre‑defined shape in this calculator and let it apply the appropriate analytical formula.

Mass Moment vs. Second Moment of Area

These two quantities are often confused. The mass moment of inertia (units $\text{kg·m}^{2}$) relates to rotational motion, while the second moment of area (units $\text{m}^{4}$) is used in beam bending and structural analysis. They represent distinct physical concepts; this tool deals exclusively with the mass moment of inertia.

Moment of Inertia Formulas for 23 Homogeneous Shapes

The following list presents the figures available in the calculator. All formulas assume uniform density and that the rotation axis passes through the centre of mass unless stated otherwise.

1. Solid sphere – axis through centre: $I = \dfrac{2}{5} m r^{2}$.
2. Thin circular hoop – axis through centre: perpendicular to plane → $I = m r^{2}$; in‑plane (x or y) → $I = \dfrac{1}{2} m r^{2}$.
3. Solid rectangular prism (cuboid) – axis through centre parallel to length ($l$): $I = \dfrac{1}{12} m (w^{2} + h^{2})$; parallel to width: $I = \dfrac{1}{12} m (l^{2} + h^{2})$; parallel to height: $I = \dfrac{1}{12} m (l^{2} + w^{2})$; along longest diagonal: $I = \dfrac{1}{12} m (l^{2} + w^{2} + h^{2})$.
4. Solid cylinder – central axis: $I = \dfrac{1}{2} m r^{2}$; transverse axis through centre: $I = \dfrac{1}{12} m (3 r^{2} + h^{2})$.
5. Cylindrical tube (hollow) – central axis: $I = \dfrac{1}{2} m (r_{1}^{2} + r_{2}^{2})$.
6. Thin cylindrical shell – central axis: $I = m r^{2}$.
7. Thin solid disk – central axis: $I = \dfrac{1}{2} m r^{2}$; in‑plane axis: $I = \dfrac{1}{4} m r^{2}$.
8. Regular dodecahedron (solid & hollow) – axis through centre and a vertex (formula provided in the tool).
9. Ellipsoid – about semi‑axis $a$: $I_{a} = \dfrac{m}{5}(b^{2}+c^{2})$; similar for $b$ and $c$.
10. Regular icosahedron (solid & hollow) – axis through centre and a vertex.
11. Isosceles triangle – axis through tip perpendicular to plane; expression depends on vertex angle and side length.
12. Regular octahedron (solid & hollow) – axis through centre and a vertex.
13. Point mass – at distance $r$: $I = m r^{2}$.
14. Thin rectangular plate – axis through centre perpendicular to plane: $I = \dfrac{1}{12} m (l^{2}+w^{2})$.
15. Regular polygon ($n$ sides, circumradius $R$) – axis through centre perpendicular to plane: $I = \dfrac{1}{2} m R^{2} \left[1 - \dfrac{2}{3} \sin^{2}\!\left(\dfrac{\pi}{n}\right)\right]$.
16. Hollow right circular cone – axes through centre (central and transverse).
17. Solid right circular cone – central axis: $I = \dfrac{3}{10} m r^{2}$; transverse axis through centre: $I = \dfrac{3}{20} m \left(r^{2} + \dfrac{h^{2}}{4}\right)$.
18. Thin rod – axis through centre perpendicular to rod: $I = \dfrac{1}{12} m L^{2}$; axis through one end: $I = \dfrac{1}{3} m L^{2}$.
19. Hollow sphere (thin shell) – axis through centre: $I = \dfrac{2}{3} m r^{2}$.
20. Spherical shell (thick) – inner radius $r_{1}$, outer radius $r_{2}$; formula in tool.
21. Regular tetrahedron (solid & hollow) – axis through centre and a vertex.
22. Torus – axis perpendicular to major diameter through centre: $I = m\left(b^{2} + \dfrac{3}{4} a^{2}\right)$; axis parallel to major diameter: $I = \dfrac{1}{2} m a^{2}$.
23. Two point masses – axis through centre of mass, perpendicular to line joining them: $I = \mu r^{2}$, where $\mu = \dfrac{m_{1} m_{2}}{m_{1}+m_{2}}$.

For any of these shapes, simply enter the requested dimensions and mass; the calculator returns the moment of inertia in the unit system you prefer.