Mass Moment of Inertia Calculator

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Calculate mass moment of inertia for common shapes and axes, or solve directly from mass and radius of gyration.

Choose whether to compute mass moment of inertia from a shape and dimensions, or compute it from mass and radius of gyration.
Pick the object shape that best matches your problem. The calculator will show only the dimensions needed for that shape.
Choose the rotation axis named for the selected shape. Use the axis wording shown in your textbook or problem statement. An axis is an imaginary line the object spins around. A centroidal axis goes through the center of mass.
Total mass of the object. Use mass, not weight. If you have a force in pounds-force (lbf), that is not mass.
All length dimensions use this unit, including radius and distance from an axis.
Use a true mass unit. Imperial mass uses slug or lbm, not lbf (force).
Distance from the center to the outer edge. Used for ring, disk, cylinder, and sphere cases.
Inside radius for a hollow cylinder or hollow sphere.
Outside radius for a hollow cylinder or hollow sphere.
Overall length of the rod, cylinder, or prism, depending on shape.
Width of a plate or prism.
Height or thickness, depending on the selected shape.
Perpendicular distance from the point mass to the rotation axis.
Equivalent distance k such that inertia I equals mass times k squared. This is called the radius of gyration.
Advanced options
Output settings
Choose the unit used for the displayed mass moment of inertia. If you pick a unit that does not match your inputs, the calculator converts using exact length and mass conversions.
Auto uses regular decimals for ordinary values and scientific notation only for very large or very small results.
Number of significant figures for scientific or engineering notation, and for auto formatting when rounding is needed.
Mass moment of inertia
Resistance to angular acceleration about the chosen axis. Larger values mean the object is harder to spin up or slow down.
Radius of gyration
Equivalent distance k where all the mass could be imagined concentrated without changing the same inertia about that axis.
Formula used
Shows which standard case was used so you can verify it matches the problem statement.
Axis summary
Helps prevent using the wrong axis for symmetric shapes.
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How to use our Mass Moment of Inertia Calculator

  1. Choose What do you want to find? to pick either Inertia from shape or Inertia from radius of gyration.
  2. If you chose shape mode, select the Shape and then the matching Axis of rotation from the problem statement.
  3. Enter Mass (selected unit), then choose Length unit and Mass unit. Use mass, not force.
  4. Fill in only the dimension fields that apply, such as Radius (selected unit), Length (selected unit), Width (selected unit), or Height (selected unit).
  5. For hollow shapes, enter both Inner radius (selected unit) and Outer radius (selected unit), making sure the outer radius is larger.
  6. If you chose gyration mode, enter Radius of gyration (selected unit) instead of shape dimensions.
  7. Open Advanced options if you want a specific Output inertia unit, a different Display format, or a custom number of Significant figures.
  8. Click Calculate to see Mass moment of inertia, Radius of gyration, the Formula used, and the Axis summary.
  9. Sanity-check the result: if the same object is moved farther from the chosen axis, inertia should be larger, and if the shown axis does not match your homework axis, change Axis of rotation before using the answer.

Definitions

Mass moment of inertia: A measure of how strongly an object's mass resists changes in rotation about a chosen axis. It depends on both mass and how far that mass is from the axis [1].

Axis of rotation: The imaginary line the object rotates around. Choosing a different axis can change the answer a lot.

Centroidal axis: An axis that passes through the object's center of mass. Many standard textbook formulas use a centroidal axis.

Radius of gyration: A distance k that gives the same inertia if the whole mass were imagined at that distance from the axis, using the relation between inertia, mass, and k [3].

Mass vs. area moment of inertia: This calculator is for rotational dynamics of mass. It is not the second moment of area used in beam bending and cross-section stiffness [3].

Common mistakes and quick fixes

Mistake: Entering weight in Mass (selected unit) instead of mass, especially using a force idea like pounds-force.
Fix: Use Mass unit with kg, g, slug, or lbm only, then enter the object's mass in Mass (selected unit) .

Mistake: Picking the wrong Axis of rotation for a symmetric shape, such as using a cylinder centerline when the problem wants a transverse axis.
Fix: Read the Axis summary after calculating and make sure it matches the exact axis named in your problem.

Mistake: Filling in hidden or irrelevant dimensions mentally and expecting them to matter for the current shape.
Fix: Only enter the visible required fields for the chosen Shape and mode; the calculator uses only those active inputs.

Mistake: Reversing Inner radius (selected unit) and Outer radius (selected unit) for a hollow cylinder or hollow sphere.
Fix: Make sure Outer radius (selected unit) is greater than Inner radius (selected unit) before you calculate.

Mistake: Mixing up Radius of gyration (selected unit) with the actual geometric radius of a disk, ring, or sphere.
Fix: Use Radius of gyration (selected unit) only in gyration mode; in shape mode, enter the real size like Radius (selected unit) .

Mistake: Comparing one Mass moment of inertia result with another result from a different axis and assuming the larger object is always harder to rotate.
Fix: Compare values only when both the object and the Axis of rotation are the same.

Limitations & Key Assumptions / Boundary Conditions

  • This tool uses standard closed-form formulas for the supported shapes and axes only. If your problem uses an offset axis, a composite body, or a custom geometry, the result will not apply unless that exact case matches one of the listed formulas.
  • Thin ring, thin disk, and thin plate cases neglect thickness. Use them only when your class or problem statement treats the object as thin.
  • Slender rod formulas assume a uniform rod and the named perpendicular axes. They are not for bent rods, tapered rods, or rods with large thickness.
  • Rectangular plate formulas assume a thin plate, while rectangular prism formulas assume a full 3D solid. Picking the wrong shape changes the result.
  • Hollow sphere here means a thin spherical shell, using the outer radius. It is not a thick-walled hollow sphere model.
  • Unit conversions are handled exactly for the listed unit pairs, but the displayed answer can still differ slightly from a textbook because of rounding and chosen significant figures.
  • Mass moment of inertia is always tied to the chosen axis. A correct shape with the wrong axis still gives the wrong answer for the real problem.

Methodology

What the calculator does

The calculator finds the mass moment of inertia for one supported rigid-body shape about one named axis, then computes the matching radius of gyration. In general, mass moment of inertia is based on how far each small piece of mass is from the axis of rotation [1].

I = integral(r^2 dm)

For the supported shapes, the calculator does not integrate directly. It uses the matching standard closed-form formula from the selected shape and axis, then converts the answer to the requested output unit.

Shape formulas used

Point mass: I = m*r^2

Thin ring or hoop, center perpendicular axis: I = m*r^2

Thin ring or hoop, center diameter: I = (1/2)*m*r^2

Solid disk, center perpendicular axis: I = (1/2)*m*r^2

Solid disk, center diameter: I = (1/4)*m*r^2

Solid cylinder, longitudinal axis: I = (1/2)*m*r^2

Solid cylinder, transverse centroidal axis: I = (1/12)*m*(3*r^2 + L^2)

Hollow cylinder, longitudinal axis: I = (1/2)*m*(r_i^2 + r_o^2)

Hollow cylinder, transverse centroidal axis: I = (1/12)*m*(3*(r_i^2 + r_o^2) + L^2)

Solid sphere, any diameter: I = (2/5)*m*r^2

Hollow sphere, any diameter: I = (2/3)*m*r^2

Slender rod, center perpendicular axis: I = (1/12)*m*L^2

Slender rod, end perpendicular axis: I = (1/3)*m*L^2

Rectangular plate, center perpendicular axis: I = (1/12)*m*(w^2 + h^2)

Rectangular plate, center x-axis: I = (1/12)*m*h^2

Rectangular plate, center y-axis: I = (1/12)*m*w^2

Rectangular prism, center x-axis: I = (1/12)*m*(w^2 + h^2)

Rectangular prism, center y-axis: I = (1/12)*m*(L^2 + h^2)

Rectangular prism, center z-axis: I = (1/12)*m*(L^2 + w^2)

Radius of gyration

After finding inertia, the calculator also reports radius of gyration. This is the equivalent distance from the axis where the full mass could be imagined to give the same inertia [3].

I = m*k^2

k = sqrt(I/m)

If you choose gyration mode, the calculator solves inertia directly from mass and radius of gyration using the same relation [3].

Unit handling

Input lengths are converted to meters using exact factors for cm, mm, in, and ft, and masses are converted to kilograms using exact factors for g and lbm, with slug converted by the stated engineering conversion set. The final inertia is then converted to the selected output unit.

Mini example

Suppose you choose a solid cylinder about the center axis along its length, with mass 10 kg and radius 0.25 m. The calculator uses the solid-cylinder longitudinal formula.

I = (1/2)*m*r^2

I = (1/2)*10*(0.25^2) = 0.3125 kg*m^2

Then it finds the radius of gyration from the inertia and mass.

k = sqrt(I/m) = sqrt(0.3125/10) = 0.1768 m

A larger k means the same mass is effectively farther from the axis, so the object is harder to speed up or slow down in rotation.

Assumptions behind the result

The formulas assume uniform mass distribution and the exact named axis for each shape. If your real object has holes, attachments, nonuniform density, or a different axis location, the textbook value and the physical value can differ.

Sources

  1. 7.4: Mass Moment of Inertia - Engineering LibreTexts - Libretexts
  2. SCTCC - PHYS 2310 Engineering Physics I Formula Sheets (PDF) - Sctcc
  3. Mass Moment of Inertia - Engineeringtoolbox