Free Moment of Inertia Calculator
Use this tool to calculate the mass moment of inertia for common shapes. Enter values in any consistent units (for example, kg and m, or lb and ft).
- Solid disk/cylinder: I = (1/2)mr²
- Hollow cylinder: I = (1/2)m(ri² + ro²)
- Solid sphere: I = (2/5)mr²
- Rod (center): I = (1/12)mL²
- Rod (end): I = (1/3)mL²
- Rectangular plate: I = (1/12)m(b² + h²)
- Point mass: I = mr²
What is moment of inertia?
Moment of inertia is a measure of how hard it is to rotate an object around a specific axis. If mass is spread farther from the axis, the moment of inertia is larger, and the object resists angular acceleration more strongly. In rotational dynamics, it plays the same role that mass plays in linear motion.
In practical terms, this means a heavy flywheel with mass concentrated near its rim is harder to spin up than one with mass near the center. Engineers, students, and hobbyists use moment of inertia in mechanics, robotics, vehicle design, machine components, and physics homework.
How to use this moment of inertia calculator for free
Step-by-step
- Select the shape and axis that matches your problem.
- Enter the mass value.
- Enter the required geometric dimensions (radius, length, width, etc.).
- Click Calculate to get the result instantly.
The result is shown in units of mass × length². If you enter kilograms and meters, output is in kg·m². If you enter pounds and feet, output is in lb·ft².
Why axis selection matters
A single object can have different moments of inertia depending on the axis of rotation. A rod rotating through its center behaves very differently from the same rod rotating about one end. This is why every formula in rotational mechanics includes a clear axis definition.
If your required axis is offset from the center of mass, use the parallel-axis theorem:
I = Icm + md²
Here, Icm is the moment of inertia about the center-of-mass axis, and d is the distance between axes.
Common formulas included in this calculator
1) Solid disk or cylinder (about central axis)
Use this for wheels, pulleys, and solid rotors. The formula is: I = (1/2)mr².
2) Hollow cylinder / thick ring
Use when an object has both inner and outer radius. Formula: I = (1/2)m(ri² + ro²).
3) Solid sphere
Useful for balls, spherical components, and model approximations: I = (2/5)mr².
4) Slender rod (center or end axis)
For a rod rotating perpendicular to its length:
- Through center: I = (1/12)mL²
- Through end: I = (1/3)mL²
5) Rectangular plate (center, perpendicular axis)
For thin flat plates: I = (1/12)m(b² + h²).
Example calculation
Suppose you have a solid disk with mass 8 kg and radius 0.3 m.
- Formula: I = (1/2)mr²
- Substitute: I = 0.5 × 8 × (0.3)²
- Result: I = 0.36 kg·m²
This value can be used directly in torque equations such as τ = Iα.
Common mistakes to avoid
- Mixing units: keep units consistent throughout input values.
- Wrong axis: always verify if the axis is through center, end, or another location.
- Using diameter as radius: many errors come from entering diameter where radius is required.
- Geometry mismatch: pick the formula that matches your physical shape assumptions.
FAQ
Is this calculator really free?
Yes. This moment of inertia calculator for free can be used without signup or payment.
Can I use centimeters instead of meters?
Yes. Use any units you like, as long as all inputs are consistent. Output units follow your chosen system.
Does this include area moment of inertia for beams?
No. This page calculates mass moment of inertia for rotational dynamics. Area moment of inertia (used in beam bending) is a different quantity.
Final thoughts
A reliable inertia estimate makes rotational design safer and more accurate. Whether you are solving classroom physics, sizing a motor, or modeling a rotating assembly, this calculator gives a fast starting point. Use it with careful unit handling and correct axis definition for the best results.