Mass Moment of Inertia Calculator finds I in kg·m² for 23 shapes, from rods and cylinders to cones and rings, about each axis, with the parallel axis offset and radius of gyration.
I = ½ m r². Axis: central axis (z).
Offset, Spin, Units
Offset feeds the parallel axis card. Speed and time feed the spin card.
Step-by-Step Calculation
- Formula, central axis (z): I = ½ m r²
- I = ½ × 10 × 0.4² = 0.8 kg·m²
- Radius of gyration: k = √(I ÷ m) = √(0.8 ÷ 10) = 0.282843 m
- Parallel axis: I at d = I about the center of mass + m × d² = 0.8 + 10 × 0.5² = 3.3 kg·m²
- Spin: ω = 10.472 rad/s, so energy = ½ × I × ω² = 43.8649 J and spin-up torque = I × ω ÷ t = 4.18879 N·m
All Axes of This Shape
| Axis | Formula | I | k |
|---|---|---|---|
| Central axis (z) | ½ m r² | 0.8 kg·m² | 28.28 cm |
| Across the middle (x, y) | 1/12 m (3 r² + h²) | 1.6 kg·m² | 40 cm |
| Across one end face | 1/12 m (3 r² + 4 h²) | 5.2 kg·m² | 72.11 cm |
Round Shapes, Same m and r
| Shape, central axis | Formula | I |
|---|---|---|
| Point mass or thin hoop | m r² | 1.6 kg·m² |
| Sphere, hollow thin wall | ⅔ m r² | 1.067 kg·m² |
| Cylinder or disk, solid | ½ m r² | 0.8 kg·m² |
| Ball, solid | ⅖ m r² | 0.64 kg·m² |
| Cone, solid | 3/10 m r² | 0.48 kg·m² |
Mass Moment of Inertia Calculator for 23 Shapes
The Mass Moment of Inertia Calculator finds how hard a body is to spin about a chosen axis. Pick one of 23 shapes, pick the rotation axis, and enter the dimensions with either the mass or the density of the material. The result comes in kg·m² or in six other units.
Explained simply, the mass moment of inertia is the rotational version of mass. Mass resists a push in a straight line. Moment of inertia resists a twist, and it depends on how far the mass sits from the axis as well as how much mass there is.
The basic formula is for a single point of mass m at a distance r from the axis. A real body is a sum of such points, and OpenStax University Physics writes that sum as an integral over the body.
$$I = m r^2$$
$$I = \int r^2 \, dm$$
Because distance is squared, the unit is mass times length squared. In SI that is kg·m², so the moment of inertia is not measured in kg or g alone. Moving the same mass twice as far from the axis makes it four times harder to spin.
Formulas for Different Shapes
Working the integral for a regular shape always gives a number times m times a length squared. These are the formulas behind each shape and axis in the Mass Moment of Inertia Calculator.
| Shape | Axis | Formula |
|---|---|---|
| Point mass | At distance r | m r² |
| Two point masses | Through the center of mass | μ r², with μ = m₁ m₂ ÷ (m₁ + m₂) |
| Rod (thin) | Through the center | 1/12 m L² |
| Rod (thin) | Through one end | ⅓ m L² |
| Circular hoop | Central axis | m r² |
| Circular hoop | Along a diameter | ½ m r² |
| Disk (thin, solid) | Central axis | ½ m r² |
| Disk (thin, solid) | Along a diameter | ¼ m r² |
| Cylinder (solid) | Central axis | ½ m r² |
| Cylinder (solid) | Across the middle | 1/12 m (3 r² + h²) |
| Cylinder (solid) | Across one end face | 1/12 m (3 r² + 4 h²) |
| Cylindrical tube (thick) | Central axis | ½ m (r₁² + r₂²) |
| Cylindrical tube (thick) | Across the middle | 1/12 m (3 (r₁² + r₂²) + h²) |
| Cylindrical shell (thin) | Central axis | m r² |
| Cylindrical shell (thin) | Across the middle | 1/12 m (6 r² + h²) |
| Ball (solid sphere) | Any axis through center | ⅖ m r² |
| Sphere (hollow, thin wall) | Any axis through center | ⅔ m r² |
| Spherical shell (thick) | Any axis through center | ⅖ m (r₂⁵ − r₁⁵) ÷ (r₂³ − r₁³) |
| Ellipsoid (solid) | About semi-axis a | 1/5 m (b² + c²) |
| Cone, solid | Central axis | 3/10 m r² |
| Cone, solid | Across the apex | 3/20 m (r² + 4 h²) |
| Cone, hollow (thin wall) | Central axis | ½ m r² |
| Cone, hollow (thin wall) | Across the apex | ¼ m (r² + 2 h²) |
| Torus (solid ring) | Central axis | ¼ m (4 R² + 3 a²) |
| Torus (solid ring) | Along a diameter | ⅛ m (4 R² + 5 a²) |
| Rectangular plate (thin) | Through center, normal | 1/12 m (l² + w²) |
| Rectangular plate (thin) | Along a long edge | ⅓ m w² |
| Cuboid (solid box) | Parallel to the height | 1/12 m (l² + w²) |
| Isosceles triangle (thin) | Through the tip, normal | ½ m L² (1 − ⅔ sin² β) |
| Triangle, base and height | Through the tip, normal | m (h² ÷ 2 + b² ÷ 24) |
| Regular polygon (thin) | Through center, normal | ½ m R² (1 − ⅔ sin²(π ÷ n)) |
| Tetrahedron | Solid, hollow | 1/20 m s², 1/12 m s² |
| Octahedron | Solid, hollow | 1/10 m s², 1/6 m s² |
| Icosahedron | Solid, hollow | φ² ÷ 10 m s², φ² ÷ 6 m s² |
| Dodecahedron | Solid, hollow | (39 φ + 28) ÷ 150 m s², (39 φ + 28) ÷ 90 m s² |
In the last rows s is the edge length and φ is the golden ratio, 1.618. The cones, the cuboid, the plate and the triangle have more axes than the table shows. The “All Axes of This Shape” panel lists every one for the shape in use, each with its own value.
The triangle has two entries for the same shape. One takes the equal side L and the tip angle 2β, and the default of 0.9 m and 40° gives 3.734 kg·m² for 10 kg. The other takes the base and the height. The regular polygon accepts its side length or its circumradius R, and a 10 kg hexagon with 0.4 m sides comes to 0.6667 kg·m².
Mass Moment of Inertia of a Cylinder
The calculator opens on a solid cylinder of 10 kg with a 0.4 m radius and a 1.2 m height. About its central axis the mass moment of inertia formula for a cylinder is ½ m r². That is ½ × 10 × 0.4², which gives 0.8 kg·m².
The height does not appear in that formula. A long cylinder and a thin disk of the same mass and radius are equally hard to spin about the central axis, so the mass moment of inertia of a disk is also 0.8 kg·m² here.
Turn the axis sideways and the height matters. Across the middle the formula is 1/12 m (3 r² + h²), which gives 1.6 kg·m². Across one end face the same cylinder reaches 5.2 kg·m², more than six times the central value.
The Step-by-Step Calculation panel writes out each line, so the Mass Moment of Inertia Calculator works with steps for any shape. It shows the unit conversions, the formula with the numbers in place, and the radius of gyration.
Hollow parts hold more inertia for the same mass. A thick tube of 10 kg with radii of 0.3 m and 0.4 m comes to 1.25 kg·m², and a thin shell of that mass and outer radius comes to 1.6 kg·m².
When the mass is not known, the Mass field can take a density. Its unit list ends with kg/m³, g/cm³, lb/ft³ and lb/in³. The Mass Moment of Inertia Calculator then works out the volume and the mass before the moment of inertia.
Make the default cylinder out of steel at 7.85 g/cm³. Its volume is 0.6032 m³, so it weighs 4,735 kg and has 378.8 kg·m² about its central axis. Density works for the 12 solid shapes. A thin or hollow shape has no volume to multiply, so it still needs its mass.
Rectangle, Plate and Box
The mass moment of inertia of a rectangle depends on which way the axis points. The Rectangular plate option in the Mass Moment of Inertia Calculator covers four axes, and the Cuboid option adds thickness and a diagonal axis.
Take a 10 kg plate that is 1.2 m long and 0.6 m wide. Spun flat like a turntable, about the axis through its center and normal to the plate, it has 1.5 kg·m². Spun about the centerline that runs along its length it has only 0.3 kg·m².
Hinged along a long edge, like a door, the same plate has 1.2 kg·m². A solid box of 1.2 by 0.6 by 0.3 m gives 0.375, 1.275 and 1.5 kg·m² about the axes parallel to its length, width and height.
Round Shapes at the Same Mass and Radius
A handy trick to remember the formulas is to rank the round shapes by their number in front of m r². The further the mass sits from the axis, the bigger the number. The “Round Shapes, Same m and r” panel shows this for the mass and radius entered.
| Shape, central axis | Formula | I for 10 kg and r = 0.4 m |
|---|---|---|
| Point mass or thin hoop | m r² | 1.6 kg·m² |
| Sphere, hollow thin wall | ⅔ m r² | 1.067 kg·m² |
| Cylinder or disk, solid | ½ m r² | 0.8 kg·m² |
| Ball, solid | ⅖ m r² | 0.64 kg·m² |
| Cone, solid | 3/10 m r² | 0.48 kg·m² |
The same rule answers the common ring question. The moment of inertia of a ring of mass 2 kg is 2 × r² about its central axis. With a 0.5 m radius that is 0.5 kg·m², and about a diameter it is half as much.
For a circle the answer depends on where the mass is. A hoop with all its mass on the rim has m r², and a filled disk has ½ m r². The second card reports this number as the Shape Factor.
Parallel Axis Theorem and Offset Axes
Every formula above is for an axis through a set point of the shape. The parallel axis theorem moves the axis sideways by a distance d and adds one term.
$$I = I_{cm} + m d^2$$
Here Icm is the value about the parallel axis through the center of mass. The fourth card of the Mass Moment of Inertia Calculator applies it to the Parallel Axis Distance. With the default 0.5 m offset the cylinder goes from 0.8 to 3.3 kg·m², and 2.5 of that comes from the offset alone.
The rod shows the theorem at work. A 10 kg rod 1.2 m long has 1.2 kg·m² about its center. Shift the axis 0.6 m to one end and it becomes 4.8 kg·m², the same as the ⅓ m L² formula.
This is also how to find the moment of inertia for any shape. Split the body into simple parts, find each part about the common axis, and add them. OpenStax does this for a 500 kg merry-go-round of 2.0 m radius with a 25 kg child at 1.0 m, and gets 1,000 + 25 = 1,025 kg·m².
The Mass Moment of Inertia Calculator works one shape at a time. Run the disk and the point mass separately, then add the two results by hand to get the total for the pair.
Units: kg·m², lb·ft² and slug·ft²
The unit for mass moment of inertia in SI is kg·m². The Result Unit list in the Mass Moment of Inertia Calculator offers six more, and the first card always shows the answer in other common units alongside the one chosen.
| Unit | Equal to 1 kg·m² |
|---|---|
| g·cm² | 10,000,000 |
| kg·cm² | 10,000 |
| lb·ft² | 23.73 |
| lb·in² | 3,417 |
| slug·ft² (lbf·ft·s²) | 0.7376 |
| lbf·in·s² | 8.851 |
The two US families differ by gravity. A value in lb·ft² uses pounds as mass. A value in slug·ft² uses pound-force, and it is the lb·ft² figure divided by 32.174. The default cylinder is 18.98 lb·ft² or 0.59 slug·ft².
Radius of Gyration, Energy and Torque
The radius of gyration k is the distance at which a single point of the same mass would have the same moment of inertia. It is the square root of I ÷ m, and for the default cylinder it is 28.28 cm.
The third card of the Mass Moment of Inertia Calculator turns the moment of inertia into motion. It uses the Spin Speed and the Spin-Up Time from the options panel.
$$KE = \frac{1}{2} I \omega^2$$
At 100 rpm the default cylinder stores 43.86 J and has an angular momentum of 8.378 kg·m²/s. Reaching that speed in 2 s takes a steady torque of 4.189 N·m, from torque = I × α.
These are the reasons to calculate a moment of inertia at all. It sizes the motor that must spin a load, and it sets how much energy a wheel stores. The Angular Acceleration Calculator works from torque and inertia, the Angular Momentum Calculator handles L, and the Flywheel Energy Storage Calculator covers stored energy and rim stress.
Mass vs Area Moment of Inertia
Two different quantities share the name. The mass moment of inertia, in kg·m², describes rotation. The area moment of inertia, also called the second moment of area, describes how a beam cross-section resists bending. Its unit is m⁴ and it has no mass in it.
The moment of inertia of an I-beam is the area kind. For a symmetric section it is [B H³ − (B − t) h³] ÷ 12, where B is the flange width, H the full depth, t the web thickness and h the clear height between flanges. An area moment of inertia calculator is the right tool for that job.
A third relative is the polar moment of area, used for shafts in torsion. The Polar Moment of Inertia Calculator covers solid and hollow shafts.
The words first and second moment come from the power of the distance. The first moment of mass is m × r, and dividing its total by the mass locates the center of mass. The second moment is m × r², which is the moment of inertia itself.
More Moment of Inertia Questions
The questions below come up around the same searches. Where an answer has numbers, they are results from the Mass Moment of Inertia Calculator at the stated entries.
Why is it called moment of inertia?
Inertia is resistance to a change in motion. A moment, in mechanics, is a quantity multiplied by a distance from an axis. Leonhard Euler introduced the term in 1765 for the resistance of a body to a change in its rotation.
What is another name for moment of inertia?
It is also called rotational inertia, angular mass, rotational mass and the second moment of mass. Engineers add the word mass, as in mass moment of inertia, to keep it apart from the area moment used for beams.
What is the difference between moment of inertia and mass?
Mass is one fixed number for a body, and it measures inertia in straight-line motion. Moment of inertia depends on the mass and on the axis. The default cylinder has a single mass of 10 kg but 0.8, 1.6 or 5.2 kg·m² about its three listed axes.
Which type of quantity is moment of inertia?
About a fixed axis it is a scalar with the dimensions mass times length squared, and it is never negative. For a body free to turn about any axis it becomes a tensor, a set of values that changes with direction.
How is moment of inertia measured?
It is measured by timing a swing. The body is hung as a pendulum or placed on a platform that twists on wires, and the period is recorded. For a twisting setup the period is 2π √(I ÷ κ), where κ is the stiffness of the wire, so I follows from the time.
What is the moment of inertia of the human body?
It changes with posture, which is how a skater controls a spin. As a rough model, a 70 kg person treated as a solid cylinder of 0.15 m radius has about 0.79 kg·m² about the head-to-toe axis. Arms held out raise the figure, and pulling them in lowers it.
How does moment of inertia apply in real life?
It shows up wherever something turns. Ten everyday examples are a flywheel, a car wheel, a bicycle wheel, a spinning skater, a diver in a tuck, a tightrope walker’s pole, a door on its hinges, a ceiling fan, a yo-yo and a playground merry-go-round.
How do you calculate inertial mass?
Inertial mass is a different idea. It is the force on a body divided by the acceleration that force produces, m = F ÷ a. It measures resistance to straight-line acceleration and needs no axis.
More tools for spinning bodies are on the Rotational Motion Calculators hub, next to the angular speed and circular motion tools.