Moment of Inertia Calculator
Choose a shape and enter its mass and size to find its moment of inertia about the standard axis.
Moment of inertia measures how hard it is to change an object's rotation. It depends on mass and how that mass is distributed from the axis. A solid disk or cylinder is ½mr², a solid sphere is ⅖mr², a hoop is mr², a rod about its center is (1/12)mL², and about its end is (1/3)mL².
How to Use This Calculator
Pick the shape you are rotating and the axis it spins about, then enter its mass in kilograms and its size — a radius for round shapes, a length for a rod. The calculator returns the moment of inertia in kilogram-metres squared.
Moment of inertia is the rotational equivalent of mass: it tells you how much torque is needed to angularly accelerate the object. The same mass resists rotation more when it sits farther from the axis.
How the Calculation Works
Every formula has the same form — mass times a characteristic distance squared, scaled by a shape factor. A hoop, with all its mass at the rim, has the largest factor (1), while a solid sphere, with mass concentrated near the center, has the smallest common factor (⅖).
That is why where the mass sits matters as much as how much there is. A rod spun about its end resists four times as much as the same rod spun about its center, because more of its mass is far from the axis.
I = k · m · r² (k depends on shape and axis)
Worked Example
For a solid disk of mass 2 kg and radius 0.5 m: I = ½ × 2 × 0.5² = ½ × 2 × 0.25 = 0.25 kg·m².
A hoop of the same mass and radius would be I = m r² = 2 × 0.25 = 0.5 kg·m² — twice as much, because all its mass is at the rim.
Assumptions and Limitations
- Assumes uniform density and the standard axis noted for each shape
- Uses SI units — mass in kg, distance in metres, result in kg·m²
- For composite objects, add the moments of inertia of the parts
- The parallel-axis theorem is needed for axes off the standard one
Who Uses a Moment of Inertia Calculator
Physics and engineering students use this to find the rotational inertia of common shapes from mass and size. Run it for mechanics homework, rotational-dynamics problems, or a quick check during a design. It applies the standard formula for the shape you choose so you can focus on the wider problem.
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Open calculator →Frequently Asked Questions
What is moment of inertia?
It is an object's resistance to changes in its rotation — the rotational analogue of mass. It depends on both the mass and how far that mass is distributed from the axis of rotation.
Why does a hoop have more inertia than a disk?
Because all a hoop's mass sits at the rim, far from the axis, while a disk spreads mass toward the center. Distance from the axis is squared in the formula, so rim mass counts most.
What is the moment of inertia of a solid sphere?
Two-fifths of mass times radius squared, ⅖mr², about an axis through its center. A hollow sphere is ⅔mr², larger because its mass is farther out.
Why is a rod's inertia different at the end?
A rod about its end is (1/3)mL², four times the (1/12)mL² about its center, because rotating from the end puts more of the rod's mass far from the axis.
What units does this use?
SI units: mass in kilograms and distance in metres, giving moment of inertia in kilogram-metres squared (kg·m²).
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