Moment of Inertia Calculator

Free Online Moment of Inertia Calculator (With Steps & Diagrams)

This moment of inertia calculator finds I — an object's rotational inertia, or its resistance to changes in spinning motion — for 10 common shapes: point masses, hoops, solid and hollow cylinders, solid and hollow spheres, rods (about the center or one end), rectangular plates, and cones. Pick a shape, enter the mass and dimensions, and the calculator instantly returns the moment of inertia, shows a labeled diagram of exactly where the axis sits, compares your shape's mass-distribution coefficient against every other shape, and works through the full formula step by step — the same way you'd solve it by hand for a physics assignment.

It also includes the Parallel Axis Theorem, so you can shift the axis away from the center of mass, and calculates the radius of gyration (k) automatically. Units are fully flexible: kilograms, grams, or pounds for mass; meters, centimeters, millimeters, feet, or inches for every dimension; and kg·m², g·cm², lb·ft², or slug·ft² for the final answer.

What Is Moment of Inertia? (Explained Simply)

Moment of inertia, usually written as I, measures how hard it is to change an object's rotational speed — how much it resists starting to spin, stopping, or speeding up. It plays the same role in rotational motion that mass plays in straight-line motion: just as a heavier object needs a bigger push to speed it up, an object with a larger moment of inertia needs a bigger torque to change how fast it spins.

But moment of inertia isn't only about how much mass an object has — it's about where that mass sits relative to the axis of rotation. Two objects with identical total mass can have very different moments of inertia if their mass is distributed differently. Mass that sits far from the axis contributes far more to I than the same amount of mass sitting close to the axis, because the contribution grows with the square of the distance.

• For a single point mass at distance r from the axis: I = m × r²
• For an extended object, I is the sum (or integral) of every tiny piece of mass multiplied by the square of its distance from the axis: I = Σ mᵢrᵢ²
• The SI unit of moment of inertia is kilogram-meter squared (kg·m²).

Moment of Inertia Formula for Common Shapes

Every rigid shape has a well-known moment of inertia formula for rotation about its natural center-of-mass axis. This calculator supports all of the formulas below — the constant in front of m·r² (or m·L²) is what changes from shape to shape, and it directly reflects how far, on average, the mass sits from the axis:

• Point mass / particle at radius r: I = m r²
• Thin hoop or ring about its central axis: I = m r²
• Solid cylinder or disk about its central axis: I = ½ m r²
• Hollow cylinder (thick ring) about its central axis: I = ½ m (r1² + r2²)
• Solid sphere about a diameter: I = (2/5) m r²
• Thin spherical shell about a diameter: I = (2/3) m r²
• Thin rod about its center: I = (1/12) m L²
• Thin rod about one end: I = (1/3) m L²
• Rectangular flat plate about the center (axis perpendicular to the face): I = (1/12) m (a² + b²)
• Solid cone about its central axis: I = (3/10) m r²

Worked Example: Moment of Inertia of a Solid Disk

Suppose a solid disk with mass 2 kg and radius 0.5 m spins about its central axis. Here's the full solution:

• Step 1 — Formula: I = ½ m r²
• Step 2 — Substitute values: I = ½ × 2 kg × (0.5 m)²
• Step 3 — Square the radius: (0.5)² = 0.25 m²
• Step 4 — Multiply: I = ½ × 2 × 0.25 = 0.25 kg·m²
• So this disk has a moment of inertia of exactly 0.25 kg·m² about its center. Try entering these exact numbers into the calculator above (or click the 'Solid disk' example button) to see the identical result and diagram generated automatically.

The Parallel Axis Theorem — Moving the Axis Off-Center

Most textbook formulas give the moment of inertia about an axis through the object's own center of mass. But in the real world, objects often rotate about a different axis — a door doesn't rotate through its center, it rotates through its hinges at the edge. The Parallel Axis Theorem handles exactly this situation: it lets you calculate the moment of inertia about any axis that is parallel to the center-of-mass axis, as long as you know the perpendicular distance d between the two axes.

• Parallel Axis Theorem: I = I(center of mass) + m d²
• Example: a rod of mass 3 kg and length 1.2 m has I(center) = (1/12) × 3 × 1.2² = 0.36 kg·m² about its middle. Its moment of inertia about one end (d = 0.6 m, half the length) is I = 0.36 + 3 × 0.6² = 0.36 + 1.08 = 1.44 kg·m² — which exactly matches the direct 'rod about one end' formula, (1/3) × 3 × 1.2² = 1.44 kg·m², confirming the theorem works.
• This calculator's Parallel Axis Theorem toggle performs exactly this calculation automatically — enter the offset distance d, and it adds m·d² to whatever center-of-mass value it just computed.

Radius of Gyration Explained

The radius of gyration, k, is a single distance that summarizes an object's entire mass distribution: it's the radius at which all the object's mass could be concentrated as a single point mass and still produce the exact same moment of inertia. It's calculated as k = √(I ÷ m), and this calculator computes it automatically for every shape and axis. Radius of gyration is especially useful in engineering — for example, in structural and mechanical design, it's used to compare how efficiently different cross-sections resist bending and buckling.

Why Mass Distribution Matters More Than Total Mass

The comparison chart above this section shows something that surprises many students the first time they see it: a hoop and a solid disk of the exact same mass and radius have very different moments of inertia — the hoop's coefficient is 1, while the disk's is only 0.5, exactly half. That's because all of the hoop's mass sits at the maximum distance r from the axis, while the disk's mass is spread out from the center all the way to the edge, with much of it sitting much closer to the axis where it contributes far less to I (since the contribution scales with distance squared, not distance). This is the same reason ice skaters spin faster when they pull their arms in — they're reducing their own moment of inertia by moving mass closer to the rotation axis, and (because angular momentum is conserved) their angular velocity has to increase to compensate.

How to Use This Moment of Inertia Calculator

Start by picking the shape and rotation axis from the dropdown — the formula and required dimensions update automatically. Enter the mass and whichever dimensions that shape needs (radius, inner/outer radius, length, or width), choosing whatever units match your problem. If your axis isn't through the center of mass, tick 'Apply Parallel Axis Theorem' and enter the offset distance d.

The results panel shows the moment of inertia, the radius of gyration, and the exact formula used. Below that, the shape diagram draws the axis, labels every dimension with your actual numbers, and displays the resulting I value directly on the picture — so the diagram alone tells the whole story. The comparison chart shows how your chosen shape's coefficient stacks up against all nine other shapes at a glance. Finally, scroll to the step-by-step solution for the complete worked-out math, exactly as you'd write it on paper for a homework assignment or exam.

Real-Life Applications of Rotational Moment of Inertia

Moment of inertia isn't just a textbook formula — it governs the behavior of a huge range of real machines, sports, vehicles, and natural systems:

• Flywheels — engines and energy-storage systems use flywheels with a deliberately large moment of inertia (mass concentrated far from the axis, like a hoop) so they resist speed changes and smooth out power delivery between engine strokes or grid demand spikes.
• Figure skating and diving — athletes pull their arms and legs in to reduce their moment of inertia and spin faster, then extend outward to slow their spin back down — a direct, visible demonstration of the mass-distribution effect this calculator quantifies.
• Vehicle wheels and tires — engineers minimize a wheel's moment of inertia (keeping mass near the hub rather than the rim) to reduce the torque needed to accelerate or brake, which is why lightweight, low-inertia wheels are prized in racing and cycling.
• Doors and hinges — a door swings about an axis through its hinges rather than its center, exactly the Parallel Axis Theorem scenario — this is why a door is much easier to push open near the handle (far from the hinge) than near the hinge itself.
• Gyroscopes and spacecraft attitude control — gyroscopes rely on a large, carefully placed moment of inertia to resist changes in orientation, which is used for navigation, image stabilization, and satellite attitude control.
• Wind turbine blades and propellers — designers balance blade moment of inertia against the torque available from wind or a motor to control how quickly the rotor can spin up or respond to gusts.
• Sports equipment — a baseball bat, golf club, or tennis racket's 'swing weight' is really its moment of inertia about the handle; manufacturers tune where mass sits along the shaft to make equipment feel lighter or more powerful to swing.
• Planetary science — a planet's moment of inertia (compared to that of a uniform sphere) reveals how its mass is distributed internally, which is one of the key ways scientists infer that Earth has a dense metallic core rather than uniform composition throughout.
• Robotics and machine design — every rotating joint or arm in a robot has a moment of inertia that engineers must calculate to correctly size motors and control acceleration, exactly using the same formulas and Parallel Axis Theorem this calculator applies.

Moment of Inertia vs. Mass: The Rotational Analogy

Moment of inertia is often called the 'rotational mass' because it plays exactly the same role in rotational equations that ordinary mass plays in linear equations. Newton's second law F = ma becomes τ = Iα (torque equals moment of inertia times angular acceleration); linear kinetic energy ½mv² becomes rotational kinetic energy ½Iω²; and linear momentum mv becomes angular momentum Iω. Understanding I as 'how much an object resists rotational acceleration,' the direct counterpart to mass resisting linear acceleration, makes every one of these rotational formulas easy to remember once the corresponding linear formula is familiar.

Tips for Solving Moment of Inertia Problems

A few habits make these problems much easier to get right on the first try:

• Always identify the shape and the exact axis first — the same object can have a very different I depending on where the axis is (through the center vs. through an edge), so double-check which case a problem is describing.
• Convert every dimension to consistent SI units (kilograms and meters) before squaring anything — squaring an un-converted unit is one of the most common sources of error.
• If the axis isn't through the center of mass, don't guess — use the Parallel Axis Theorem (I = I(cm) + m d²) rather than trying to derive a new formula from scratch.
• Remember that distance from the axis matters far more than total mass — doubling an object's radius while keeping mass fixed quadruples I, because r is squared in every formula.
• Use the radius of gyration as a sanity check: if k comes out larger than the object's own physical radius or length, that's a sign of a units or arithmetic mistake.

Frequently Asked Questions

What is the formula for moment of inertia? It depends on the shape and axis — for a point mass it's I = m r²; for a solid disk about its center it's I = ½ m r²; for a solid sphere about a diameter it's I = (2/5) m r². This calculator includes all 10 standard shapes.

What is the SI unit of moment of inertia? The SI unit is kilogram-meter squared (kg·m²).

How do you calculate moment of inertia for an off-center axis? Use the Parallel Axis Theorem: I = I(center of mass) + m d², where d is the distance between the new axis and the center-of-mass axis. This calculator's toggle applies it automatically.

Why does moment of inertia depend on where the mass is located? Because each piece of mass contributes m × r² to the total, where r is its own distance from the axis — mass far from the axis contributes far more than the same mass close to the axis, since distance is squared.

What is the difference between moment of inertia and mass? Mass measures resistance to linear acceleration and is a single fixed property of an object. Moment of inertia measures resistance to rotational (angular) acceleration and depends on both the mass and how that mass is distributed relative to a specific axis — the same object can have several different moments of inertia depending on which axis it rotates about.

What is the radius of gyration? It's the distance k = √(I ÷ m) at which all of an object's mass could be concentrated as a single point and produce the same moment of inertia — this calculator computes it automatically for every shape.