Moment of Inertia Calculator
Calculate area moment of inertia for cross-sections or mass moment of inertia for rotating bodies using common engineering formulas.
Calculator is for informational purposes only. Terms and Conditions
Choose the inertia type and shape
Area moment uses length⁴ units for cross-sections. Mass moment uses mass × length² units for rotating bodies.
Enter the known values
Only inputs required for the selected inertia type and shape are active.
Visual Check
Use the diagram to verify the selected shape, dimensions, and axis.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See the equation, unit conversions, substitutions, assumptions, and result path
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Source/standard information updates based on the selected inertia type and shape.
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Moment of Inertia Calculator
The Moment of Inertia Calculator above calculates either area moment of inertia for a cross-section or mass moment of inertia for a rotating body. Use area moment for beam bending, deflection, and section stiffness. Use mass moment for torque, angular acceleration, flywheels, disks, rods, cylinders, rings, and other rotating bodies.
The most important step is choosing the correct inertia type. Area moment of inertia uses units like \(mm^4\), \(in^4\), or \(m^4\). Mass moment of inertia uses units like \(kg \cdot m^2\), \(slug \cdot ft^2\), or \(lbm \cdot ft^2\). These are not interchangeable because they describe different physical properties.
Quick Answer
To calculate area moment of inertia, select a cross-section shape, choose the axis, and enter the dimensions. To calculate mass moment of inertia, select a rotating body and enter its mass and radius, diameter, or length. A larger moment of inertia means the area or mass is distributed farther from the selected axis, making the section harder to bend or the body harder to angularly accelerate.
Do not rely on a simplified calculator when…
Do not use this calculator alone for final structural design, machine design, product certification, or code compliance. Real designs may require material properties, load combinations, section modulus, deflection limits, buckling checks, local slenderness, welds, holes, fillets, bearing conditions, dynamic effects, and professional engineering review.
Inputs and Outputs Used by the Calculator
The calculator inputs change based on whether you are solving for area moment of inertia or mass moment of inertia. The output also changes units because the two calculations describe different physical properties.
| Type | Value | What It Means | Common Unit |
|---|---|---|---|
| Input | Inertia Type | Selects whether the calculation is for cross-section bending or rotating-body dynamics. | Area or Mass |
| Input | Shape | Defines which standard formula applies, such as rectangle, centered tube, pipe, symmetric triangle, ideal I-beam, disk, rod, ring, or sphere. | Shape selection |
| Input | Axis | Defines whether the result is about the x-axis, y-axis, center axis, end axis, or another listed axis. | \(x\), \(y\), center, end |
| Input | Dimensions | Width, height, diameter, inner diameter, length, flange size, or web thickness depending on the shape. | mm, m, in, ft |
| Input | Mass | Mass of a rotating body when using mass moment of inertia mode. | kg, slug, lbm |
| Input | Offset Distance | Distance from the centroidal axis to a parallel target axis if using the parallel axis theorem. | mm, m, in, ft |
| Output | Area Moment of Inertia | Geometric stiffness property used in beam bending, deflection, and section property checks. | \(mm^4\), \(in^4\), \(m^4\) |
| Output | Mass Moment of Inertia | Rotational inertia used with torque, angular acceleration, and rotational kinetic energy. | \(kg \cdot m^2\), \(slug \cdot ft^2\) |
Moment of Inertia Formulas
The formula depends on the type of inertia, the shape, and the axis. The calculator uses standard centroidal formulas for common idealized shapes and applies the parallel axis theorem when an offset axis is selected.
Rectangle Area Moment of Inertia
Use this for a rectangular cross-section about its centroidal x-axis. The height term is cubed, so increasing section depth has a large effect on \(I_x\).
Circle Area Moment of Inertia
For a solid circle about a centroidal diameter, \(I_x=I_y\). For a pipe, subtract the inner diameter term from the outer diameter term.
Parallel Axis Theorem
Use this when the desired area moment of inertia is about an axis parallel to the centroidal axis but offset by distance \(d\).
Mass Moment of Inertia for a Solid Disk or Cylinder
Use this for a solid disk or cylinder rotating about its central symmetry axis.
Area Moment of Inertia Formulas by Shape
The formulas below are for idealized centroidal axes unless noted otherwise. The calculator’s parallel axis option can shift an area moment of inertia to a parallel offset axis.
| Shape | Formula | Assumption |
|---|---|---|
| Rectangle | \(I_x=\frac{b h^3}{12}\), \(I_y=\frac{h b^3}{12}\) | Centroidal x-axis and y-axis. |
| Centered Hollow Rectangle / Tube | \(I_x=\frac{b h^3-b_i h_i^3}{12}\), \(I_y=\frac{h b^3-h_i b_i^3}{12}\) | Inner rectangular void is centered within the outer rectangle. |
| Solid Circle | \(I=\frac{\pi D^4}{64}\) | Centroidal diametral axis. \(I_x=I_y\). |
| Hollow Circle / Pipe | \(I=\frac{\pi(D^4-d^4)}{64}\) | Concentric circular tube. \(D\) is outer diameter and \(d\) is inner diameter. |
| Symmetric Triangle | \(I_x=\frac{b h^3}{36}\), \(I_y=\frac{h b^3}{48}\) | Assumes an isosceles or symmetric triangle with centroidal axes aligned to the symmetry axis. |
| Symmetric I-Beam | \(I_x=2\left(\frac{b_f t_f^3}{12}+b_f t_f y_f^2\right)+\frac{t_w h_w^3}{12}\) | Idealized rectangular flanges and web. No fillets, tapers, holes, welds, or rolled-shape radii. |
| Parallel Axis Theorem | \(I=I_c+A d^2\) | Target axis must be parallel to the centroidal axis. |
Mass Moment of Inertia Formulas by Body
The formulas below are for ideal rigid bodies rotating about the listed axis. For mass moment calculations, enter mass, not weight or force.
| Body | Formula | Axis |
|---|---|---|
| Solid Disk / Solid Cylinder | \(I=\frac{1}{2}m r^2\) | Central symmetry axis. |
| Hollow Cylinder | \(I=\frac{1}{2}m(r_o^2+r_i^2)\) | Central symmetry axis. |
| Thin Ring | \(I=m r^2\) | Central axis perpendicular to ring plane. |
| Slender Rod About Center | \(I=\frac{1}{12}m L^2\) | Axis through center, perpendicular to rod. |
| Slender Rod About End | \(I=\frac{1}{3}m L^2\) | Axis through one end, perpendicular to rod. |
| Solid Sphere | \(I=\frac{2}{5}m r^2\) | Axis through any diameter. |
Mass parallel axis note
A similar parallel axis theorem exists for mass moment of inertia: \(I=I_{cm}+md^2\). The calculator’s offset-axis option is intended for area moment of inertia unless a dedicated mass-offset mode is provided.
Practical insight
Moment of inertia is dominated by distance from the axis. For area moment, moving material farther from the neutral axis often increases bending stiffness more efficiently than adding material near the center.
What the Variables Mean
Every variable must match the selected formula and unit system. For area moment calculations, dimensions control the result. For mass moment calculations, both mass and distance from the rotation axis matter.
| Symbol | Meaning | How to Enter It |
|---|---|---|
| \(I\) | Moment of inertia about the selected axis. | Output value calculated by the tool. |
| \(I_x\) | Area moment of inertia about the x-axis. | Select x-axis or strong-axis direction when applicable. |
| \(I_y\) | Area moment of inertia about the y-axis. | Select y-axis or weak-axis direction when applicable. |
| \(I_c\) | Centroidal moment of inertia. | Used before applying an offset-axis correction. |
| \(A\) | Cross-sectional area. | Computed from the selected shape and dimensions. |
| \(d\) | Offset distance between parallel axes. | Enter only when using the parallel axis theorem. |
| \(b\) | Width or base of a cross-section. | Enter as a positive dimension with units. |
| \(h\) | Height or depth of a cross-section. | Enter as a positive dimension with units. |
| \(D\) | Outer diameter. | Used for circles, pipes, disks, cylinders, rings, and spheres. |
| \(r\) | Radius from the rotation axis. | Often calculated as \(r=D/2\). |
| \(m\) | Mass of a rotating body. | Enter mass, not weight or force. |
How to Use the Calculator
Start by deciding whether your problem is a beam or cross-section problem, or a rotating-body problem. That decision determines the units, formula, and interpretation of the result.
Choose area or mass moment
Select Area Moment of Inertia for cross-sections and bending. Select Mass Moment of Inertia for rotating bodies.
Select the shape
Choose the closest ideal shape, such as rectangle, centered hollow rectangle, circle, pipe, symmetric triangle, symmetric I-beam, disk, hollow cylinder, ring, rod, or sphere.
Confirm the axis
For cross-sections, choose \(I_x\) or \(I_y\). For rotating bodies, verify that the listed axis matches the real rotation axis.
Enter dimensions and units
Use consistent dimensions. For hollow shapes, the inner dimension must be smaller than the outer dimension.
Review the result and checks
Check the output units, quick checks, solution steps, and any warnings before using the result in another calculation.
Which inertia type should you choose?
| Your Problem | Use This | Typical Units |
|---|---|---|
| Beam bending or deflection | Area moment of inertia | \(mm^4\), \(in^4\) |
| Torque and angular acceleration | Mass moment of inertia | \(kg \cdot m^2\), \(slug \cdot ft^2\) |
| Torsion of a circular shaft | Polar area moment of inertia | \(mm^4\), \(in^4\) |
| Bending stress | Area moment first, then section modulus | \(I\) in \(mm^4\), \(S\) in \(mm^3\) |
| Column slenderness or buckling | Area moment and radius of gyration | \(I\), \(A\), and \(r=\sqrt{I/A}\) |
How to Interpret the Result
A larger moment of inertia means more area or mass is distributed farther from the selected axis. For beams, that usually means more bending stiffness. For rotating bodies, it means more torque is required for the same angular acceleration.
| Result Pattern | What It May Mean | What to Check Next |
|---|---|---|
| Large \(I_x\) compared with \(I_y\) | The section is much stiffer about one bending axis than the other. | Check whether the strong axis or weak axis is being used for the load direction. |
| Very small area inertia | The section may be thin, shallow, or entered in the wrong units. | Confirm mm vs m, inches vs feet, and shape dimensions. |
| Large mass inertia | Mass is far from the rotation axis or the mass value is large. | Check whether the value entered is mass, not weight or force. |
| Offset inertia much larger than centroidal inertia | The parallel axis term \(Ad^2\) dominates the result. | Verify the offset distance and confirm the axis is parallel. |
| Impossible or negative result | Invalid geometry or incorrectly entered hollow dimensions. | Check that inner dimensions are smaller than outer dimensions. |
What to do with the result
Use area moment of inertia in beam deflection, bending stress, section modulus, and radius of gyration checks. Use mass moment of inertia in torque, angular acceleration, flywheel, shaft, and rotating body calculations.
What changes the result most?
Distance from the axis usually changes the result most. In a rectangle, \(I_x=\frac{bh^3}{12}\), so doubling height increases \(I_x\) by a factor of eight if width stays constant. For rotating bodies, doubling radius increases mass moment of inertia by a factor of four if mass stays constant.
Quick sanity check
Area moment of inertia must use length to the fourth power, such as \(in^4\) or \(mm^4\). Mass moment of inertia must use mass times length squared, such as \(kg \cdot m^2\). If your output unit does not match the physical problem, you selected the wrong inertia type.
Input Quality Checklist
Most wrong moment of inertia results come from the wrong type, wrong axis, or wrong units. Use this checklist before trusting the output.
Confirm the physical problem
Use area moment for cross-sections and beam behavior. Use mass moment for rotating bodies.
Check the axis
Make sure the selected axis matches the real bending axis or rotation axis.
Verify hollow dimensions
Inner width, inner height, or inner diameter must be smaller than the corresponding outer dimension.
Use mass, not force
For mass moment of inertia, enter mass in kg, slug, or lbm. Do not enter weight in newtons or pounds-force.
Watch fourth-power units
Area moment unit conversions are not linear. A small length-unit mistake becomes a large inertia error.
Use ideal shapes carefully
Fillets, holes, welds, rounded corners, tapered flanges, and cutouts can change the exact section properties.
Step-by-Step Worked Examples
Moment of inertia has mixed search intent, so it helps to see both area moment and mass moment examples. The examples below match common calculator use cases.
Formula
Substitution
Calculation
Result
\(I_x \approx 66.7 \times 10^6\,mm^4\). This is reasonable because the section is twice as tall as it is wide, so the x-axis inertia is much larger than the y-axis inertia.
Formula
Substitution
Calculation
Result
\(I \approx 4.27 \times 10^6\,mm^4\). The hollow center removes material near the centroid, where it contributes less to bending resistance than material farther away.
Formula
Substitution
Calculation
Result
\(I=0.0125\,kg \cdot m^2\). This value can be used with \(\tau=I\alpha\) if the disk rotates about the same central axis used in the formula.
Why height, radius, and offset distance matter
In area moment formulas, dimensions often appear to the third or fourth power. In mass moment formulas, distance from the rotation axis is squared. That is why shape orientation, radius, and offset distance can dominate the final result.
Moment of Inertia Concept Diagram
Moment of inertia depends on how far area or mass is distributed from the selected axis. The simple in-page diagram below shows the key idea without requiring a separate image file.
Reference Values and Reasonableness Checks
There is no universal “good” moment of inertia because the correct value depends on the shape, size, axis, load case, material, and design goal. However, the patterns below help identify suspicious results.
| Situation | Expected Pattern | Suspicious Result |
|---|---|---|
| Tall rectangular section | \(I_x\) should usually be much larger than \(I_y\) when height is greater than width. | \(I_x\) and \(I_y\) are swapped or nearly equal for a tall rectangle. |
| Solid circle | \(I_x=I_y\) about centroidal diameters. | Different \(I_x\) and \(I_y\) values for a centered solid circle. |
| Hollow pipe | Inertia should be less than the solid outer circle but greater than a very thin solid core. | Inner diameter is larger than outer diameter or result is negative. |
| Parallel axis theorem | Offset inertia should be greater than or equal to centroidal inertia. | Offset inertia is smaller than centroidal inertia. |
| Mass moment of disk | Increasing radius should strongly increase \(I\). | Radius or diameter was entered with the wrong units. |
Design Ranges and Practical Engineering Checks
Moment of inertia is usually an input to another engineering check, not the final design answer. A high inertia value may reduce deflection or angular acceleration, but final suitability depends on the full system.
Beam stiffness
For the same material and span, a larger area moment of inertia generally reduces bending deflection.
Axis sensitivity
A section can be strong about one axis and weak about another. Always match \(I_x\) or \(I_y\) to the bending direction.
Rotational response
A larger mass moment of inertia requires more torque to reach the same angular acceleration.
Engineering judgment check
For structural use, also check bending stress, shear, deflection, lateral stability, local buckling, supports, loads, material properties, and code requirements. For rotating machinery, also check speed, torque, bearings, balance, fatigue, startup conditions, and safety factors.
Moment of Inertia Units
Unit consistency is critical because area moment uses length to the fourth power, while mass moment uses mass times length squared. Mixing these unit systems is one of the most common causes of wrong results.
| Quantity | Common Units | Conversion Reminder |
|---|---|---|
| Area Moment of Inertia | \(mm^4\), \(cm^4\), \(m^4\), \(in^4\), \(ft^4\) | \(1\,in^4=416{,}231.4256\,mm^4\) |
| Area Moment of Inertia | \(ft^4\) and \(in^4\) | \(1\,ft^4=20{,}736\,in^4\) |
| Area Moment of Inertia | \(m^4\) and \(mm^4\) | \(1\,m^4=10^{12}\,mm^4\) |
| Mass Moment of Inertia | \(kg \cdot m^2\) | SI unit for rotational inertia. |
| Mass Moment of Inertia | \(slug \cdot ft^2\) | Useful U.S. engineering unit when torque is in \(lbf \cdot ft\). |
| Mass Moment of Inertia | \(lbm \cdot ft^2\) | Mass-based unit that must not be confused with pound-force. |
Hidden unit trap
Do not convert area moment of inertia by converting the numeric value only once. Because units are length to the fourth power, the length conversion factor must be raised to the fourth power. That is why an inch-to-millimeter mistake can create a result that is off by hundreds of thousands.
Area Moment vs. Mass Moment vs. Polar Moment
Similar names can describe different properties. The safest approach is to identify the physical problem first, then choose the matching formula and units.
| Term | Used For | Typical Units | Related Formula |
|---|---|---|---|
| Area Moment of Inertia | Beam bending, deflection, section stiffness | \(mm^4\), \(in^4\) | \(I_x=\frac{b h^3}{12}\) |
| Mass Moment of Inertia | Rotational dynamics, torque, angular acceleration | \(kg \cdot m^2\), \(slug \cdot ft^2\) | \(I=\frac{1}{2}mr^2\) |
| Polar Area Moment | Torsion and circular shaft behavior | \(mm^4\), \(in^4\) | \(J=I_x+I_y\) |
| Section Modulus | Bending stress checks | \(mm^3\), \(in^3\) | \(S=\frac{I}{c}\) |
| Radius of Gyration | Column slenderness and buckling checks | mm, in, m | \(r=\sqrt{\frac{I}{A}}\) |
Common Mistakes That Cause Wrong Results
Moment of inertia formulas are sensitive to axis choice, geometry, and units. These mistakes can create results that look precise but are not physically useful.
Common Mistakes
- Using area moment of inertia when the problem requires mass moment of inertia.
- Using \(I_x\) when the load bends the section about the y-axis.
- Entering diameter where the formula expects radius.
- Entering weight or force instead of mass for rotating bodies.
- Forgetting that \(mm^4\), \(in^4\), and \(m^4\) convert by the fourth power.
- Using centroidal formulas for an offset axis without applying \(I=I_c+Ad^2\).
Better Practice
- Start by identifying whether the problem is bending or rotation.
- Draw the axis before selecting \(I_x\), \(I_y\), or a rotation formula.
- Use consistent units for every dimension.
- Check inner dimensions before calculating hollow shapes.
- Apply the parallel axis theorem when shifting away from the centroidal axis.
- Use the result as an input to stress, deflection, torque, or buckling checks.
Troubleshooting Unexpected Results
If the result looks wrong, check the type, axis, and units before changing the formula. Most errors come from setup, not arithmetic.
| Problem | Likely Cause | Fix |
|---|---|---|
| Result unit is \(kg \cdot m^2\) when you expected \(in^4\) | Mass moment mode was selected instead of area moment mode. | Switch to area moment of inertia for beam or cross-section calculations. |
| Result is negative | Inner dimension is larger than the outer dimension for a hollow shape. | Verify inside width, inside height, or inside diameter. |
| \(I_x\) seems too small for a tall beam | Axis may be swapped or width and height may be reversed. | Check the axis diagram and confirm which dimension is depth. |
| Parallel-axis result is unexpectedly huge | Offset distance is large or entered in the wrong unit. | Confirm offset distance and units before using the result. |
| Mass inertia seems too high | Diameter may have been entered as radius or weight was entered as mass. | Use the correct radius/diameter convention and enter mass only. |
Edge case to watch
Very thin-walled shapes can be sensitive to small dimension errors. A small mistake in wall thickness can noticeably change area, radius of gyration, and section properties.
Assumptions, Sources, and Limitations
This calculator is intended for educational use, preliminary checks, and quick engineering estimates. It uses idealized geometry and standard rigid-body or section-property formulas.
Ideal Geometry
Shapes are treated as perfect rectangles, centered rectangular tubes, concentric pipes, symmetric triangles, idealized I-beams, disks, cylinders, rods, rings, or spheres.
Centroidal Axis Basis
Standard area formulas are centroidal unless the parallel axis theorem option is applied.
Rigid Body Basis
Mass moment formulas assume rigid bodies with ideal mass distribution around the selected axis.
Design Limit
The calculator does not check code compliance, yielding, buckling, fatigue, welds, bearings, supports, or manufacturer limits.
Calculation basis
The formulas in this guide are standard engineering mechanics relationships for second moments of area, rigid-body mass moments of inertia, and the parallel axis theorem. For additional background on area moment of inertia and centroidal-axis methods, see the University of Alberta engineering statics notes on moments of inertia of area.
Final design caution
For structural or mechanical design, use the calculated inertia value only as one part of the design process. Final decisions should account for real geometry, rounded corners, fillets, bolt holes, welds, tapered flanges, composite materials, cracked sections, local buckling, warping torsion, nonuniform density, dynamic imbalance, bearing friction, loading, material behavior, applicable standards, safety factors, and professional engineering judgment.
Glossary of Terms
These terms explain the most important ideas behind the calculator.
Moment of Inertia
A property that describes how area or mass is distributed around an axis.
Area Moment of Inertia
A geometric property of a cross-section used in bending, deflection, and section-property calculations.
Mass Moment of Inertia
A rotational property that describes how difficult it is to angularly accelerate a body about an axis.
Centroidal Axis
An axis that passes through the centroid of a shape.
Parallel Axis Theorem
A method for shifting centroidal area inertia to a parallel offset axis using \(I=I_c+Ad^2\).
Radius of Gyration
A distance that relates area to moment of inertia using \(r=\sqrt{I/A}\).
Polar Moment of Inertia
A section property used in torsion, commonly calculated as \(J=I_x+I_y\) for an area.
Strong Axis
The bending axis with the larger area moment of inertia for a given cross-section.
Frequently Asked Questions
What does the Moment of Inertia Calculator calculate?
The calculator finds either area moment of inertia for cross-sections or mass moment of inertia for rotating bodies. Area moment of inertia is used for beam bending and deflection. Mass moment of inertia is used for torque, angular acceleration, and rotational motion.
What is the difference between area moment of inertia and mass moment of inertia?
Area moment of inertia depends only on cross-section geometry and uses length-to-the-fourth-power units such as \(mm^4\) or \(in^4\). Mass moment of inertia depends on mass and distance from the rotation axis and uses mass-times-length-squared units such as \(kg \cdot m^2\) or \(slug \cdot ft^2\).
What units should I use for moment of inertia?
Use \(mm^4\), \(cm^4\), \(m^4\), \(in^4\), or \(ft^4\) for area moment of inertia. Use \(kg \cdot m^2\), \(slug \cdot ft^2\), or \(lbm \cdot ft^2\) for mass moment of inertia. Do not mix area moment units with mass moment units.
Why do Ix and Iy give different results?
\(I_x\) and \(I_y\) are measured about different axes. If more area is distributed far from the x-axis than the y-axis, \(I_x\) will be larger. Axis direction matters because distance from the axis strongly affects the inertia value.
When should I use the parallel axis theorem?
Use the parallel axis theorem when the desired area moment of inertia is about an axis that is parallel to, but offset from, the centroidal axis. The formula is \(I=I_c+Ad^2\), where \(I_c\) is centroidal inertia, \(A\) is area, and \(d\) is the offset distance.
Can this calculator be used for final structural or mechanical design?
The calculator can support educational checks and preliminary calculations, but final structural or mechanical design should also consider codes, loads, material properties, manufacturer data, safety factors, and professional engineering judgment.