What is the formula for a rod rotating about its center?

I = (1/12) × M × L², where M is the mass in kg and L is the length in meters. The result is in kg·m².

What is the formula for a rod rotating about one end?

I = (1/3) × M × L². This is exactly four times larger than the end-to-center formula, which follows directly from the parallel axis theorem.

Why is the moment of inertia larger when rotating about the end?

Mass farther from the axis contributes more (proportional to r²). Rotating about the end places the far half of the rod at greater distances, increasing the total inertia.

What units does this calculator use?

Mass in kilograms (kg), length in meters (m), and the result in kilogram-square-meters (kg·m²). Convert your inputs before entering them if you are using other units.

Does this apply to non-uniform rods?

No. The formulas I = (1/12)ML² and I = (1/3)ML² assume the density is uniform along the entire length. For a non-uniform rod you must integrate ρ(x)x² dx over the length.

How do I use this for a beam pinned at one end (pendulum)?

Select "One End" as the axis. This gives I = (1/3)ML², which is the correct value for a physical pendulum or a rod swinging about a pivot at its tip.

Moment of Inertia Rod

Enter the mass and length of a uniform rod to calculate its moment of inertia about its center or one end.

Rod Parameters

Mass (M) — kg

Length (L) — m

Rotation Axis

Center
I = ML²/12 One End
I = ML²/3

Result

Enter mass and length, then click Calculate.

Formula Reference

I = (1/12) × M × L²

Axis through the center of the rod, perpendicular to its length.

I = (1/3) × M × L²

Axis through one end of the rod, perpendicular to its length.

M = mass in kg · L = length in m · I in kg·m²

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Summary

The moment of inertia of a uniform rod measures its resistance to rotational acceleration about a given axis. For an axis through the center, I = (1/12)ML², and for an axis through one end, I = (1/3)ML², where M is the total mass and L is the rod length. These values are derived using the parallel axis theorem.

How it works

Enter the total mass of the rod in kilograms.
Enter the total length of the rod in meters.
Select whether the rotation axis passes through the center or through one end.
The calculator instantly displays the moment of inertia in kg·m².
Use the formula breakdown to understand how the result was computed.

Use cases

Mechanical engineering design of rotating shafts and beams.
Physics coursework and homework involving rigid body dynamics.
Robotics: computing torque requirements for arm segments.
Structural analysis of beams under angular acceleration.
Comparing rotational resistance for different rod configurations.
Verifying textbook results against hand calculations.

Frequently Asked Questions

Last updated: 2026-06-11 · Reviewed by Nham Vu