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Walk-through

How to Use This Calculator

3 steps

1

Pick a shape and enter mass and radius/length

On the By Shape tab, choose the object's shape — point mass, rod, disk, sphere, hoop, or cylinder — then enter its mass and characteristic radius or length.

2

Read the moment of inertia

The result card shows I in kg·m² along with the exact formula used, so you can see which numbers drove the answer.

3

Shift the axis or compare shapes

Switch to Parallel Axis to move the rotation axis off-center, or Compare to see how I changes across all eight shapes at the same mass and radius.

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Reference

Formula & Methodology

7 formulas

Point mass

I = m r^2

A single mass m orbiting at a fixed distance r from the axis. Every other shape's formula is built from summing this expression over its mass distribution.

Thin rod

I = (1/12) m L^2 (about center) | I = (1/3) m L^2 (about end)

A uniform rod of length L and mass m spins four times harder about one end than about its center, because more of its mass sits far from an end axis.

Solid disk or cylinder

I = (1/2) m r^2

A uniform disk or cylinder of radius r spinning about its central symmetry axis. Mass is spread from the center out to r, not concentrated at the rim.

Solid sphere

I = (2/5) m r^2

A uniform solid ball of radius r spinning through its center. Most of a sphere's mass sits closer to the axis than a disk's does, so its coefficient is smaller.

Hollow sphere (thin shell)

I = (2/3) m r^2

A thin spherical shell of radius r, with essentially all mass at distance r from the center, so it resists rotation more than a solid sphere of the same mass.

Hoop or thin ring

I = m r^2

A thin ring of radius r with all mass concentrated at the rim — the largest coefficient of any shape at a given mass and radius.

Parallel-axis theorem

I = I_cm + m d^2

The moment of inertia about any axis parallel to one through the center of mass equals I_cm plus m times the square of the offset distance d between the two axes.

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Glossary

Key Terms Explained

7 terms

Moment of inertia (I) A measure of how much an object resists a change in its rotational speed, in kg·m². It plays the same role for rotation that mass plays for straight-line motion.

Rotational inertia Another name for moment of inertia — the resistance a spinning body offers to a change in its angular velocity when a torque is applied.

Axis of rotation The imaginary line an object spins around. The same object has a different moment of inertia for every axis you choose.

Parallel-axis theorem A shortcut formula, I = I_cm + m d^2, that finds I about any axis parallel to the center-of-mass axis without re-deriving the integral from scratch.

Radius of gyration The distance k from the axis at which all of an object's mass could be concentrated to produce the same I, defined by I = m k^2.

Mass distribution How an object's mass is spread relative to the rotation axis. Two objects with equal mass and equal outer radius can have very different I if their mass sits nearer to or farther from the axis.

Angular acceleration The rate of change of angular velocity, related to torque and moment of inertia by τ = I α — the rotational counterpart to F = m a.

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Scenarios

Real-World Examples

3 worked examples

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Physics student

Solid disk on a spin table

Shape Solid disk Mass 2 kg Radius 0.5 m

I = (1/2)(2)(0.5)^2 = 0.25 kg·m². Because the disk's mass is spread from the center outward rather than concentrated at the rim, it resists spin-up less than a hoop of the same mass and radius would.

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Astronomy hobbyist

Modeling a planet as a solid sphere

Shape Solid sphere Mass 5 kg (scaled model) Radius 2 m

I = (2/5)(5)(2)^2 = 8 kg·m². The 2/5 coefficient is the smallest of the common shapes because a sphere's mass is, on average, closer to its own center than a disk's or a shell's mass is.

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Mechanical engineer

Flywheel offset from its mounting axis

Shape Solid disk, Icm = 0.25 kg·m² Mass 2 kg Axis offset (d) 1 m

I = Icm + m d^2 = 0.25 + (2)(1)^2 = 2.25 kg·m². Moving the rotation axis just 1 m off-center increased the moment of inertia ninefold, showing how sensitive I is to axis placement, not just shape.

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Deep Dive

Understanding the Moment of Inertia Calculator

3 chapters · 4 min read

Moment of inertia (I) tells you how hard it is to speed up or slow down an object's spin, the same way ordinary mass tells you how hard it is to speed up or slow it down in a straight line. This calculator works out I for the seven most common textbook shapes, plus how I changes when the rotation axis is shifted off the center of mass.

How the Moment of Inertia Calculator works

Each shape's formula comes from integrating r² over every bit of mass in the object, where r is that bit's distance from the axis. A point mass concentrates everything at one distance, giving the simplest case, I = m r². Extended shapes — rods, disks, spheres, hoops — spread mass across a range of distances, so their formulas carry a fractional coefficient (1/12, 1/3, 1/2, 2/5, 2/3, or 1) that captures how that mass is distributed relative to r or L. These are standard rigid-body mechanics results for uniform-density objects about the axis named for each shape.

Inputs and what they mean

Mass (m) is the object's total mass in kilograms. The radius or length input (r or L) is the object's characteristic size — the radius for disks, spheres, hoops, cylinders, and a point mass's distance from the axis, or the full length for a rod. On the Parallel Axis tab, the axis offset (d) is the distance between the object's own center-of-mass axis and the new axis you actually want I about; a larger offset increases I by m d², regardless of the object's shape.

Limits and edge cases

These formulas assume uniform density and idealized shapes — a real flywheel with spokes, a non-uniform sphere, or a rod with end weights will have a different I than the simple textbook formula gives. The calculator also assumes SI units throughout (kilograms, meters, kg·m²); mixing unit systems will silently produce a wrong answer. For a torque-driven spin-up calculation once you have I, see the Torque Calculator and the relation τ = I α.

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Questions

Frequently Asked Questions

6 questions

What does the Moment of Inertia Calculator actually compute?+

It computes I, the rotational equivalent of mass, for a chosen shape (point mass, rod, disk, sphere, hoop, or cylinder) given that shape's mass and characteristic radius or length. I is measured in kilogram-meters-squared (kg·m²) and tells you how much torque it takes to change the object's spin rate.

Why is the disk formula I = 1/2 m r² but the hoop is I = m r²?+

Both have the same mass and radius, but a hoop concentrates all of its mass at the rim (distance r from the axis), while a disk spreads its mass evenly from the center out to r. Because moment of inertia weights mass by the square of its distance from the axis, the hoop's mass-at-the-rim resists spinning twice as much as the disk's spread-out mass.

What is the parallel-axis theorem and when do I need it?+

The parallel-axis theorem, I = I_cm + m d², lets you find the moment of inertia about any axis that runs parallel to (but offset by distance d from) an object's center-of-mass axis. You need it whenever the actual rotation axis — a hinge, a mounting bolt, a pivot point — isn't through the object's own center of mass.

Why do a solid sphere and a hollow sphere give different results for the same mass and radius?+

A solid sphere's mass is distributed throughout its volume, so most of it sits closer to the center than the outer radius r. A hollow sphere (thin shell) has essentially all of its mass at distance r. Since I grows with the square of distance from the axis, the hollow sphere's coefficient (2/3) is larger than the solid sphere's (2/5) even though both share the same m and r.

What units does the calculator expect?+

Mass in kilograms (kg) and radius or length in meters (m). The result, I, comes out in kilogram-meters-squared (kg·m²), the standard SI unit for moment of inertia. If your measurements are in other units, convert them to kg and m before entering them.

Can I share my results?+

Yes — the calculator encodes your selected tab, shape, mass, radius/length, and (on the Parallel Axis tab) axis offset into the page URL. Copying the URL, or clicking the Share button, is enough to hand a specific scenario to someone else.

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Moment of Inertia Calculator

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