Calculate the moment of inertia (rotational inertia) of a solid or hollow cylinder, sphere, or rod from its mass and dimensions, plus its radius of gyration.
Results
Calculated
Moment of inertia, I
—
kg·m² about the chosen axis
Radius of gyration, k
—
k = √(I / M), in meters
Formula used
—
Coefficient c in I = c × M × R² or L²
Dimension used
—
Radius or length applied in the formula (m)
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How to use this calculator
Pick the shape and rotation axis that matches your object, then enter its mass and the relevant dimension (radius for cylinders and spheres, length for rods). Click Calculate to get the moment of inertia, radius of gyration, and the formula that was applied. Click Clear to reset all fields and start a new calculation.
The formulas used
Each shape below uses its standard textbook formula for moment of inertia I about the stated axis, where M is mass, R is radius, and L is length:
Solid cylinder or disk (central axis): I = (1/2) M R²
Thin hollow cylinder / hoop (central axis): I = M R²
Solid sphere (about a diameter): I = (2/5) M R²
Thin hollow sphere (about a diameter): I = (2/3) M R²
Thin rod (about its center, perpendicular to its length): I = (1/12) M L²
Thin rod (about one end, perpendicular to its length): I = (1/3) M L²
Understanding the inputs
Mass must be in kilograms and the radius or length in meters for the moment of inertia to come out in the SI unit kg·m². Only the dimension relevant to your chosen shape is used in the calculation — radius for cylinders and spheres, length for rods — so you can leave the other field at its default without affecting the result.
Interpreting the results
The moment of inertia I is the primary output: a larger I means more torque is needed to produce the same angular acceleration. The radius of gyration k = √(I / M) restates that same resistance as an equivalent distance from the axis, which is useful for comparing shapes of different mass on equal footing. The formula and dimension shown let you verify which coefficient and length were actually used.
Frequently Asked Questions
What is moment of inertia?
Moment of inertia (also called rotational inertia), I, measures how much an object resists changes to its rotational speed about a given axis. It plays the same role in rotational motion that mass plays in straight-line motion: torque = I × angular acceleration, just as force = mass × acceleration for linear motion. It depends on both the mass and how that mass is distributed relative to the rotation axis.
Why does the same object have a different moment of inertia for different shapes?
Moment of inertia depends on how far the mass sits from the rotation axis, not just on total mass. A hollow cylinder (I = MR²) has all its mass concentrated at radius R, while a solid cylinder of the same mass and radius (I = (1/2)MR²) has mass distributed all the way in to the center, so it resists rotation less. This is why solid and hollow shapes, and rods rotated about the center versus about one end, use different coefficients.
What is radius of gyration?
The radius of gyration k is the distance from the axis at which the object's entire mass could be concentrated as a single point and still produce the same moment of inertia: k = √(I / M). It is a convenient way to compare the mass distribution of different shapes independent of their total mass.
Do these formulas apply to real, non-uniform objects?
These formulas assume idealized rigid bodies with uniform density and the exact geometry named (a truly thin-walled hoop, a perfectly solid sphere, and so on). Real objects with holes, varying density, or added parts deviate from these ideals, so treat the result as a close approximation and refine it with the parallel axis theorem or numerical integration when precision matters.
Practical Guide for Moment of Inertia Calculator | Rotational Inertia for Common Shapes
Moment of Inertia Calculator | Rotational Inertia for Common Shapes is most useful when the inputs reflect the situation you are actually planning around, not a best-case estimate. Treat the result as a decision aid: it gives you a structured way to compare assumptions, spot outliers, and decide what to verify next. For Physics work, the most important review lens is units, idealized assumptions, boundary conditions, measurement precision, and expected physical scale.
Start with a baseline run using values you can defend. Then change one assumption at a time and watch which output moves the most. If one input dominates the result, spend your verification time there first. If several inputs have similar influence, use a conservative scenario and an optimistic scenario to create a practical range instead of relying on a single exact number.
Before acting on the result, verify the output with dimensional analysis, known reference values, or a second formula when possible. This is especially important when the calculator supports a purchase, project plan, performance target, or operational decision. The calculator can make the math consistent, but the quality of the conclusion still depends on current data, clear units, and assumptions that match your real constraints.
When the output looks surprising, slow down and inspect each input in order. A small change in one high-leverage field can move the final number more than several low-leverage fields combined. For Moment of Inertia Calculator | Rotational Inertia for Common Shapes, that means you should first confirm the value with the greatest scale, then confirm the value with the greatest uncertainty, then rerun the calculator with conservative and optimistic assumptions. This sequence turns the calculator from a single answer into a practical decision range.
Review Checklist
Confirm every input uses the unit and time period requested by the calculator.
Run a low, expected, and high scenario so the answer has a useful range.
Check whether rounding or a missing decimal place changes the decision.
Update the calculation whenever the object, medium, force, distance, time, or measurement method changes.