What is the difference between mass moment of inertia and area moment of inertia?

Mass moment of inertia (kg·m²) measures resistance to angular acceleration and appears in τ = Iα. Area moment of inertia (m⁴) describes how a cross-section resists bending and appears in σ = My/I and beam deflection formulas. They share a name and a shape of formula but are different quantities with different units.

What is the parallel axis theorem?

I = Icm + m d². It gives the moment of inertia about any axis parallel to one through the center of mass, where d is the distance between the two axes. The center-of-mass value is always the smallest, so shifting the axis always increases I.

How do I calculate the energy stored in a flywheel?

Find the moment of inertia I about the spin axis, convert speed to rad/s with ω = 2πn/60, then compute KE = ½Iω². A 10 kg solid disc of 0.15 m radius at 3000 rpm stores about 5.55 kJ.

How much torque do I need to accelerate a rotating mass?

τ = Iα, where α is the angular acceleration in rad/s². To reach ω in time t from rest, α = ω/t. Add the torque needed to overcome friction and load, because this calculator reports the inertial part only.

Why does a hoop have a larger moment of inertia than a disc of the same mass and radius?

All of a hoop's mass sits at the full radius, giving mr². A disc spreads its mass over smaller radii on average, giving ½mr². Inertia depends on how far the mass is from the axis, and it grows with the square of that distance.

Mass Moment of Inertia Calculator

Mass moment of inertia of common solids, the parallel axis theorem, flywheel rotational kinetic energy, and the torque needed for a given angular acceleration.

Scratchpad (not saved)

Save to create a persistent, shareable workspace.

Shape

Mass m

Total mass of the body.

Mass Unit

Unit for the value entered above.

Dimension a

Radius r for discs, cylinders, rings, and spheres (outer radius r₂ for a hollow cylinder); length L for a rod; side a for a plate or box.

Dimension b

Inner radius r₁ for a hollow cylinder; height h for a transverse cylinder; side b for a plate or box. Ignored for other shapes.

Dimension Unit

Unit for the value entered above.

Axis Offset d (optional)

Distance from the center-of-mass axis to the parallel axis of rotation. Use 0 to rotate about the center. Kinetic energy and torque use the inertia about this axis.

Speed n (optional)

rpm

For rotational kinetic energy. Use 0 to skip.

Angular Acceleration α (optional)

rad/s²

For the torque τ = Iα. Use 0 to skip.

What This Calculator Does

Mass moment of inertia, in kg·m², measures how strongly a body resists changes in its rotation, and it governs flywheels, rotors, wheels, and anything you spin up or brake. It is a different quantity from the area moment of inertia in m⁴, which describes how a cross-section resists bending and is what the moment of inertia calculator computes for beams. This calculator covers eight common solids, shifts the axis with the parallel axis theorem, and from the result gives rotational kinetic energy at a chosen speed and the torque needed for a chosen angular acceleration.

It combines Shape, Mass m, Mass Unit, Dimension a to estimate Mass moment of inertia about the center, Icm, Rotational kinetic energy ½Iω², Angular speed ω.

Formula & Method

Rotational inertia about the center of mass, by shape: solid cylinder or disc

I_{cm} = \tfrac{1}{2}mr^2

; thick hollow cylinder

I_{cm} = \tfrac{1}{2}m\left(r_1^2 + r_2^2\right)

; thin ring

I_{cm} = mr^2

; solid cylinder, transverse axis

I_{cm} = \tfrac{1}{12}m\left(3r^2 + h^2\right)

; solid sphere

I_{cm} = \tfrac{2}{5}mr^2

; thin spherical shell

I_{cm} = \tfrac{2}{3}mr^2

; slender rod

I_{cm} = \tfrac{1}{12}mL^2

; rectangular plate

I_{cm} = \tfrac{1}{12}m\left(a^2 + b^2\right)

. Parallel axis theorem:

I = I_{cm} + m\,d^2

Rotational kinetic energy:

KE = \tfrac{1}{2} I \omega^2, \qquad \omega = \frac{2\pi n}{60}

Torque for angular acceleration:

\tau = I\,\alpha

where

m

is mass,

d

is the distance from the center of mass to the new axis,

n

is speed in rpm, and

\alpha

is angular acceleration in rad/s².

Notation used in the formulas: $R$ = Mass moment of inertia about the center, Icm; $x_{1}$ = Shape; $x_{2}$ = Mass m; $x_{3}$ = Mass Unit; $x_{4}$ = Dimension a; $x_{5}$ = Dimension b; $x_{6}$ = Dimension Unit.

Method summary: inputs are normalized to consistent units, core equations are evaluated, then secondary values are derived and rounded for display.

Use it to size a flywheel for stored energy, estimate the torque a motor needs to spin up a rotor in a given time, or find the inertia of a part about an axis that does not pass through its center, such as a rod pivoting at one end.

Worked Examples

Solid disc flywheel at 3000 rpm (metric)

Given: m = 10 kg, r = 0.15 m, n = 3000 rpm, axis through the center Method: solid disc about its axis, so I = ½mr². Then convert rpm to rad/s for the energy. I = 0.5 × 10 × 0.15² = 0.1125 kg·m² ω = 2π × 3000 / 60 = 314.16 rad/s KE = ½ × 0.1125 × 314.16² = 5551.7 J Spinning it up from rest in 2 s needs α = 157.08 rad/s², so τ = 0.1125 × 157.08 = 17.67 N·m.

Slender rod pivoted at one end (imperial)

Given: m = 5 lb, L = 3 ft, pivot at one end, α = 10 rad/s² Method: find I about the center, then shift the axis by d = L/2 with the parallel axis theorem. I_cm = (1/12) × 5 × 3² = 3.75 lb·ft² I_end = 3.75 + 5 × 1.5² = 15.0 lb·ft² (this equals mL²/3, as expected) In SI, I_end = 0.632 kg·m², so τ = 0.632 × 10 = 6.32 N·m = 4.66 lbf·ft.

Mass vs. Area Moment of Inertia

	Mass moment of inertia	Area moment of inertia
Symbol	$I$ or $J$	$I_x$, $I_y$
Units	kg·m² (lb·ft²)	m⁴ (in⁴)
Describes	Resistance to angular acceleration	Resistance of a cross-section to bending
Depends on	Mass distribution	Shape of the cross-section only
Used in	$\tau = I\alpha$, flywheels, rotor dynamics	$\sigma = My/I$, beam deflection

Reference

The shape formulas are standard results found in Meriam and Kraige, Engineering Mechanics: Dynamics, and in Hibbeler's Engineering Mechanics: Dynamics.

Common Mistakes

Using the area moment of inertia where the mass moment is needed, or the reverse. Their units differ (m⁴ against kg·m²), so a unit check catches it.
Applying the parallel axis theorem to an axis that does not start from the center of mass. $I_{cm}$ must be the starting point.
Entering a diameter in a radius field. The fields here take radius for round shapes.
Mixing up the axis. A cylinder spinning about its length has $\tfrac{1}{2}mr^2$, but the same cylinder tumbling end over end uses the transverse formula.
Forgetting to convert rpm to rad/s before computing energy or torque. This calculator does it, but a hand calculation often does not.

When Not To Use This Calculator

Bodies with non-uniform density, holes, or attachments. The formulas assume uniform density.
Composite bodies. Compute each simple part about its own center, shift each with the parallel axis theorem, and add them.
Rotation about an axis that is tilted relative to the principal axes, which needs the full inertia tensor.
Beam bending or deflection. Use the area moment of inertia calculator.

Frequently Asked Questions

What is the difference between mass moment of inertia and area moment of inertia?: Mass moment of inertia (kg·m²) measures resistance to angular acceleration and appears in τ = Iα. Area moment of inertia (m⁴) describes how a cross-section resists bending and appears in σ = My/I and beam deflection formulas. They share a name and a shape of formula but are different quantities with different units.
What is the parallel axis theorem?: I = Icm + m d². It gives the moment of inertia about any axis parallel to one through the center of mass, where d is the distance between the two axes. The center-of-mass value is always the smallest, so shifting the axis always increases I.
How do I calculate the energy stored in a flywheel?: Find the moment of inertia I about the spin axis, convert speed to rad/s with ω = 2πn/60, then compute KE = ½Iω². A 10 kg solid disc of 0.15 m radius at 3000 rpm stores about 5.55 kJ.
How much torque do I need to accelerate a rotating mass?: τ = Iα, where α is the angular acceleration in rad/s². To reach ω in time t from rest, α = ω/t. Add the torque needed to overcome friction and load, because this calculator reports the inertial part only.
Why does a hoop have a larger moment of inertia than a disc of the same mass and radius?: All of a hoop's mass sits at the full radius, giving mr². A disc spreads its mass over smaller radii on average, giving ½mr². Inertia depends on how far the mass is from the axis, and it grows with the square of that distance.

Inputs Used

Shape: Used directly in the calculation.
Mass m: Total mass of the body.
Mass Unit: Unit for the value entered above.
Dimension a: Radius r for discs, cylinders, rings, and spheres (outer radius r₂ for a hollow cylinder); length L for a rod; side a for a plate or box.
Dimension b: Inner radius r₁ for a hollow cylinder; height h for a transverse cylinder; side b for a plate or box. Ignored for other shapes.
Dimension Unit: Unit for the value entered above.
Axis Offset d (optional): Distance from the center-of-mass axis to the parallel axis of rotation. Use 0 to rotate about the center. Kinetic energy and torque use the inertia about this axis.
Speed n (optional): For rotational kinetic energy. Use 0 to skip.
Angular Acceleration α (optional): For the torque τ = Iα. Use 0 to skip.

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