Moment of Inertia Calculator
Calculate rotational inertia for common shapes, then adjust the result with the parallel-axis theorem when the rotation axis is offset.
Rotational inertia for common shapes
A moment of inertia calculator finds how strongly an object resists angular acceleration about a chosen axis. The result depends on mass, geometry, and how far the mass is distributed from the rotation axis.
A moment of inertia calculator calculates an object's resistance to angular acceleration based on mass distribution and axis of rotation. Common formulas include I = mr^2 for a point mass, I = 1/2mr^2 for a solid disk, and I = 1/12mL^2 for a thin rod rotating about its center.
Choose mass moment mode for rotating bodies or area moment mode for beam cross-sections. The calculator can return mass moment of inertia, area moments Ix and Iy, polar moment J, centroidal values, and parallel-axis adjustments.
Use consistent units. If mass is in kilograms and length is in meters, mass moment is kg*m^2. Area moment of inertia uses length^4, such as m^4, cm^4, or in^4.
Moment of inertia
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Total inertia
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Center-axis inertia
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Parallel-axis term
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Radius of gyration
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Equivalent distance k = sqrt(I / m)
Step-by-step work
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Note: Moment of inertia depends on the exact axis. A disk rotating around its central axle, a disk rotating around a diameter, and a disk rotating around an offset axis have different results.
How to use the moment of inertia calculator
- Choose the calculation type: Use mass moment for rotating bodies, or area moment for beam and cross-section geometry.
- Choose the shape: Select the idealized object, section, and axis that best match your problem.
- Enter the required values: For mass moment, enter mass and dimensions. For area moment, enter cross-section dimensions.
- Add an axis offset if needed: In mass mode, enter d. In area mode, enter dx and dy for shifted section axes.
- Read the result: The output shows the relevant base value, offset term, final result, and formula steps.
- Check the diagram: Use the preview to confirm that the selected shape and axis match the intended setup.
Moment of inertia formulas
Moment of inertia measures how mass is distributed relative to a rotation axis. Mass farther from the axis contributes more because distance is squared.
Point mass: I = m r^2
Rigid body: I = integral r^2 dm
Parallel axis theorem: I = I_cm + m d^2
The formulas in this calculator assume uniform density and standard textbook axes. For composite objects, calculate each part about the same axis and add the moments together.
Concept reference: NASA STEMonstrations - Moment of Inertia.
Common moment of inertia formulas
| Shape | Axis | Formula | Inputs |
|---|---|---|---|
| Point mass | Distance r from axis | I = m r^2 | mass, distance |
| Solid disk or cylinder | Central symmetry axis | I = (1/2) m r^2 | mass, radius |
| Thin ring or hoop | Central symmetry axis | I = m r^2 | mass, radius |
| Thick-walled cylinder | Central symmetry axis | I = (1/2) m (r1^2 + r2^2) | mass, inner radius, outer radius |
| Slender rod | Center or end axis | I = (1/12) m L^2 or (1/3) m L^2 | mass, length |
| Rectangular plate | Center axis perpendicular to face | I = (1/12) m (w^2 + h^2) | mass, width, height |
When to use the parallel-axis theorem
The parallel-axis theorem applies when you already know the moment of inertia about a center-of-mass axis and need the inertia about a new axis that is parallel to it. The offset distance d is the shortest distance between the two axes.
Use it
The new axis is parallel to the center axis, such as a disk spinning about an axle shifted sideways.
Do not use it
The new axis is tilted, angled, or completely different from the listed formula axis.
Check d carefully
Because d is squared, a small measurement error in offset distance can noticeably change the final result.
Parallel-axis reference: Oregon State University - The Parallel Axis Theorem.
Radius of gyration
Radius of gyration turns a distributed object into an equivalent point-mass distance. If the full mass were placed at distance k from the axis, it would have the same moment of inertia.
k = sqrt(I / m)
A larger k means more of the mass is effectively farther from the axis. That is why a ring and a solid disk with the same mass and radius do not have the same rotational inertia.
Measurement checks before using the answer
Most moment of inertia mistakes come from selecting the wrong axis, mixing diameter with radius, or combining units accidentally.
Radius vs diameter
If a problem gives diameter, divide by 2 before entering radius.
Uniform density
The listed formulas assume mass is evenly distributed within the idealized shape.
Composite parts
Break complex objects into simple pieces, move each piece to the same axis, then add the results.
Mass moment, area moment, and polar moment are different
Many search results and textbook problems use the words "moment of inertia" for different properties. Before copying a result, check whether the problem is asking about rotational motion, beam bending, or torsion.
| Property | What it uses | Typical unit | Used for |
|---|---|---|---|
| Mass moment of inertia | Mass distribution around an axis | kg*m^2 or lb*in^2 | Rotation, torque, angular acceleration, flywheels, pulleys |
| Area moment of inertia | Cross-section geometry around a centroidal axis | m^4 or in^4 | Beam bending stress, deflection, section stiffness |
| Polar moment of area | Cross-section area distribution around a polar axis | m^4 or in^4 | Torsion, shaft twist, circular section checks |
This calculator returns mass moment of inertia. If your problem mentions a beam, cross-section, deflection, section modulus, or centroidal area property, you likely need an area moment of inertia calculator instead.
Area moment reference: Mississippi State University - Second Moments of Area.
Area moment formulas included in this calculator
Area mode calculates centroidal Ix and Iy first, then applies Ix = Ix,c + A dy^2 and Iy = Iy,c + A dx^2 if you enter offsets. The polar moment shown in the result is J = Ix + Iy for the selected section axes.
| Cross-section | Ix,c | Iy,c | Notes |
|---|---|---|---|
| Rectangle | b h^3 / 12 | h b^3 / 12 | Use width b and height h. |
| Circle | pi r^4 / 4 | pi r^4 / 4 | Same about any centroidal diameter. |
| Annulus | pi (R^4 - r^4) / 4 | pi (R^4 - r^4) / 4 | Outer radius R must be larger than inner radius r. |
| Centered triangle | b h^3 / 36 | h b^3 / 48 | Assumes the vertex is centered above the base. |
| Ellipse | pi a b^3 / 4 | pi b a^3 / 4 | a is horizontal radius, b is vertical radius. |
Worked examples with quick sanity checks
These examples show the scale of the result and help catch common mistakes like entering diameter as radius or forgetting an offset axis.
Solid disk
10 kg, radius 0.5 m
I = 1/2mr^2 = 1/2 x 10 x 0.5^2 = 1.25 kg*m^2. A thin ring with the same mass and radius would be 2.5 kg*m^2.
Center rod
3 kg, length 2 m
I = 1/12mL^2 = 1/12 x 3 x 2^2 = 1 kg*m^2. Around one end, the same rod becomes 4 kg*m^2.
Offset axis
Disk shifted by 0.2 m
For the disk above, add m d^2 = 10 x 0.2^2 = 0.4 kg*m^2. Total I = 1.25 + 0.4 = 1.65 kg*m^2.
Composite object workflow
For assemblies, brackets, wheels with hubs, or parts made from several simple shapes, the most reliable hand method is to put every part on the same final axis before adding the results.
Step 1
Break it apart
Model the object as disks, rods, rings, plates, point masses, or other simple pieces.
Step 2
Find each centroid
Measure the distance from each part's centroidal axis to the final rotation axis.
Step 3
Shift each part
Use I = I_cm + m d^2 for every part whose own axis is parallel but offset.
Step 4
Add totals
Sum every adjusted inertia value. Subtract holes only if you are using an appropriate mass or density model.
Interesting fact
A thin hoop has a moment of inertia of MR^2 about its central axis, while a solid disk with the same mass and radius has 1/2MR^2. That means the hoop's rotational inertia is 100% larger than the disk's, even though both objects have the same mass and outside radius. The difference comes from mass distribution: the hoop keeps nearly all of its mass at the outer radius, while the disk spreads mass from the center outward. Source: OpenStax Physics, Table A20 Moments of Inertia.
Frequently Asked Questions
What does a moment of inertia calculator measure?
A moment of inertia calculator measures rotational inertia about a specific axis. In this page, the main result is mass moment of inertia, which tells how strongly a mass distribution resists angular acceleration during rotation. A larger moment of inertia means more torque is needed to create the same angular acceleration.
Why do the rotation axis and centroid matter so much?
The same object can have different moments of inertia about different axes because the distances from each small mass element to the axis change. A rod spinning around its centroid has a smaller moment than the same rod spinning around one end. When the new axis is parallel to the centroidal axis, the parallel axis theorem adjusts the result with the m d^2 term.
Is this mass moment of inertia or area moment of inertia?
This calculator focuses on mass moment of inertia, which uses mass and length units such as kg*m^2. Area moment of inertia is different: it describes a cross-section, such as a rectangle, circle, or beam section, and is commonly used for bending stress and deflection calculations. Polar moment is another section property used for twisting or torsion, so it should not be mixed with the mass-based result here.
Can I use this for a real machine part with nonuniform density?
You can use it for estimates and checks, but real machine parts may have holes, nonuniform density, complex geometry, assemblies, and nonstandard axes. If density changes across the shape, the mass distribution changes too, which changes the moment of inertia. For design-critical work, verify the result with CAD mass properties, engineering calculations, or a qualified professional.
How do I calculate a composite shape's moment of inertia?
Split the object into simple shapes, calculate each part's moment of inertia about the same final axis, and add the values. For example, a composite part may be modeled as a rectangle plus a circle, or a beam-like section may be broken into smaller cross-section pieces. If a part's own centroid axis is not the final axis, use the parallel-axis theorem before adding it to the total.
Why is a ring's moment of inertia larger than a disk's?
A ring places almost all of its mass at the outer radius, while a solid disk spreads mass from the center to the edge. Because the formula weights distance from the axis by r^2, mass near the outside contributes much more. That is why a thin ring uses I = m r^2, while a solid disk uses I = 1/2 m r^2 for the same rotation axis.
What should I enter if a circle problem gives diameter?
Enter radius, not diameter, for circular shapes unless the input field specifically says otherwise. Divide the diameter by 2 before using the calculator. This matters because radius is squared in many moment of inertia formulas, so entering diameter by mistake can make the result four times too large.
Does the parallel axis theorem work for any new axis?
No. The parallel axis theorem only works when the new axis is parallel to the center-of-mass or centroid axis used by the formula. If the new axis is tilted, rotated, or not parallel, the calculation needs a different inertia formula or a full tensor approach.
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Disclaimer: This moment of inertia calculator is for general educational and informational use only. It uses simplified ideal-shape formulas and user-entered values, and it may not represent real parts with holes, variable density, complex assemblies, welds, fasteners, bearings, deformations, or nonstandard axes. It is not a substitute for professional engineering analysis, safety review, finite element analysis, CAD mass-property verification, or manufacturer specifications. Always verify design-critical, structural, mechanical, aerospace, automotive, robotics, medical, or safety-related calculations with a qualified professional before building, purchasing, operating, or relying on equipment.
Last updated: May 21, 2026