How To Calculate Moment Of Inertia | Stop Guessing, Get It Right

Moment of inertia sounds like a single number you can “look up,” until the axis changes and your answer flips. That’s the part that trips people.

This page makes it simple. You’ll learn one core rule that always works, then you’ll use shortcuts for the shapes you see in real problems.

What Moment Of Inertia Means In Plain Terms

Moment of inertia tells you how hard it is to spin an object around a chosen axis. Put more mass farther from that axis and the object fights changes in spin more.

The axis is not decoration. Change the axis and you change the distances, so the number changes too.

Mass Moment Vs. Area Moment

“Moment of inertia” can mean two related things, and the units tell you which one you’re using.

Mass moment of inertia shows up in dynamics and rotation. It uses kilograms and meters: kg·m².

Area moment of inertia (second moment of area) shows up in beam bending and stiffness. It uses length to the fourth power: m⁴ or in⁴.

Start With The Core Rule

For a set of point masses, the rule is direct: add up each mass times the square of its perpendicular distance to the axis.

Written compactly: I = Σ(m·r²). The distance r is always the shortest distance from the mass to the axis line.

How To Calculate Moment Of Inertia Step By Step

Use this same sequence every time. It keeps you from mixing up axes, distances, and units.

Step 1: Name The Axis Out Loud

Write the axis in words, not just a letter. “About the center, through the shaft” is different from “about the edge” even if both are labeled z in a sketch.

If the axis is tilted or offset, draw it as a line, not a dot.

Step 2: Break The Object Into Pieces You Can Measure

If you have a few separate masses, treat them as point masses. If you have a solid shape, treat it as many tiny pieces of mass spread out continuously.

Either way, you’re still adding m·r². The only change is whether you add a list or add an integral.

Step 3: Write r As A Real Distance, Not A Vibe

r is a perpendicular distance to the axis. For a point in the xy-plane with a z-axis through the origin, r is the planar distance √(x² + y²).

For an axis that’s shifted, r often becomes √((x − a)² + (y − b)²). That shift is the whole game.

Step 4: Add It Up Or Integrate It

Discrete masses: use Σ(m·r²). Continuous mass: use I = ∫ r² dm.

To use the integral, you need dm written in terms of density and a tiny length/area/volume element.

Step 5: Check Units And Sanity

For mass moment of inertia, the unit must be mass times length squared. If you get kg·m or kg·m³, something went sideways.

Also do a quick reality check: pushing mass outward should raise I, pulling mass inward should drop I.

Turning The Integral Into Something You Can Compute

Most textbook shapes are solved once and reused forever. Still, it helps to see how the integral is built so you can adapt it.

Pick The Right Density Model

If the object is a thin rod or wire, you often use linear density: dm = λ dx.

If it’s a thin plate, use area density: dm = σ dA. For a solid, use volume density: dm = ρ dV.

Choose A Coordinate System That Matches The Shape

Round things are happiest in polar or cylindrical coordinates. Boxes and plates are happiest in x-y coordinates.

This is not about style. A friendly coordinate choice turns r² into something clean and keeps the integral short.

Worked Example: Three Point Masses On A Line

Say you have three masses on the x-axis: 2 kg at x = 0 m, 1 kg at x = 0.5 m, and 3 kg at x = 1.0 m. You want I about an axis through x = 0, perpendicular to the line (so r is just the x-distance).

Compute: I = 2·0² + 1·(0.5)² + 3·(1.0)² = 0 + 0.25 + 3 = 3.25 kg·m².

Now shift the axis to x = 1.0 m. Distances become 1.0 m, 0.5 m, and 0 m, so I = 2·1² + 1·(0.5)² + 3·0² = 2 + 0.25 + 0 = 2.25 kg·m².

Calculating The Moment Of Inertia For Common Shapes

If your object matches a standard shape, use the standard formula first. Then adjust for a shifted axis with a theorem, instead of redoing integrals from scratch.

These formulas assume uniform density unless noted.

When A Table Formula Does Not Match Your Axis

Most mistakes happen here: you grab the correct formula for the shape, then you quietly switch the axis in your head.

If your axis is parallel to a standard axis but shifted, use the parallel axis theorem. If the axis lies in a flat plate setup, the perpendicular axis theorem may help too.

If you want a deeper walk-through of rigid-body mass moments (including axis choices and symmetry rules), MIT’s Engineering Dynamics materials are a solid reference. The lecture page on mass moment topics is here: MIT OCW lecture on mass moment of inertia.

Parallel Axis Theorem Without The Hand-Waving

The parallel axis theorem lets you move from a center-of-mass axis to another axis that’s parallel to it.

The formula is: I = I_cm + m d², where d is the distance between the two parallel axes.

What d Means In Real Geometry

d is the perpendicular distance between the axes, not between two points. If the new axis is shifted sideways by 0.2 m, then d = 0.2 m even if the object is tall or wide.

This is also why the theorem is fast: you only need one distance and the total mass.

Worked Example: Rod About The End Using A Center Formula

Take a thin rod of mass 4 kg and length 0.8 m. You know the center-axis formula: I_cm = (1/12) m L².

Compute I_cm = (1/12)·4·(0.8)² = (1/12)·4·0.64 = 0.2133 kg·m².

The end axis is parallel to the center axis, shifted by d = L/2 = 0.4 m. Add m d² = 4·(0.4)² = 4·0.16 = 0.64.

So I = 0.2133 + 0.64 = 0.8533 kg·m². That matches the standard end formula (1/3) m L² when you compute it directly.

Perpendicular Axis Theorem For Flat Shapes

This one is for a thin lamina (a flat plate) in the xy-plane. If you want the moment about the z-axis through the same point, you can add the two in-plane moments.

The rule is: I_z = I_x + I_y, with all axes meeting at the same point and z perpendicular to the plate.

When It’s Safe To Use

It works for thin plates where thickness does not matter in the mass distribution for the axis you care about. If the object is a chunky 3D solid, don’t force this rule onto it.

If your sketch is clearly a “flat sheet” problem, this theorem is a clean shortcut.

Moment Of Inertia With Composite Objects

Real objects are often a mash-up of simple shapes: a hub plus a rim, a bracket plus a stiffener, a bar with holes.

The nice part is that moment of inertia adds. You can split the object into parts, find each part’s I about the same axis, then sum them.

Handling Cutouts And Holes

A hole is a “negative mass” region in the math. Find the moment for the full shape, then subtract the moment for the removed shape about the same axis.

Be strict about axes. If the removed shape’s center is offset, apply the parallel axis theorem to that removed piece before subtracting.

Worked Example: Disk With An Off-Center Hole

Take a solid disk of mass 6 kg and radius 0.3 m. A circular hole is drilled out with radius 0.1 m whose center is 0.15 m from the disk center. Treat the removed piece as a smaller disk with mass proportional to area.

Area ratio is (0.1²)/(0.3²) = 0.01/0.09 = 1/9, so removed mass is 6·(1/9) = 0.6667 kg.

Original disk I about center axis: (1/2)·6·(0.3)² = 0.5·6·0.09 = 0.27 kg·m².

Removed disk I about its own center axis: (1/2)·0.6667·(0.1)² = 0.5·0.6667·0.01 = 0.003333 kg·m².

Shift that removed disk to the big disk’s axis: add m d² = 0.6667·(0.15)² = 0.6667·0.0225 = 0.015.

Removed piece about the big axis is 0.003333 + 0.015 = 0.018333. Subtract: 0.27 − 0.018333 = 0.251667 kg·m².

Radius Of Gyration As A Clean Summary Number

Sometimes you want one distance that captures “how far out” the mass sits from the axis. Radius of gyration does that.

It’s defined by I = m k². That means k = √(I/m).

If two objects share the same mass, the one with the larger k has more mass sitting farther from the axis.

What Changes When The Axis Tilts In 3D

In full 3D rotation, a single scalar moment of inertia is not always enough. A rigid body can need a matrix (the inertia tensor) to track rotation about different axes through the same point.

Still, a lot of homework and engineering checks reduce to one axis at a time, which is why the scalar formulas and theorems get so much mileage.

Using Symmetry To Save Time

If a body is symmetric about an axis, cross-terms drop out and the principal axes line up with that symmetry. That can turn a messy setup into a clean one.

When in doubt, pick axes through the center of mass and aligned with symmetry first.

A Quick Reality Test You Can Do Before You Submit

Try a thought check with the same mass: a hoop and a solid disk of the same radius. The hoop should have the larger moment of inertia because more of its mass sits at the outer radius.

Also test axis shifts: moving the axis away from the center should raise I, since average distance squared rises.

Seeing Moment Of Inertia In Action

If you like a visual that sticks, NASA has a short demonstration showing how pulling mass inward changes rotation rate. It’s a clean way to connect the math to what your eyes see.

You can watch it here: NASA STEMonstrations on moment of inertia.

Putting It All Together On Your Next Problem

Start by naming the axis in words. Then pick your approach: a table formula, a sum of point masses, or an integral with dm.

If the axis is shifted, use the parallel axis theorem and keep d as an axis-to-axis distance. If it’s a flat plate and you need the perpendicular axis, add the two in-plane moments.

Finish by checking units and doing a fast sanity test: more mass farther out means a larger result. Do that, and your moment of inertia work stops feeling like guesswork.