Two different quantities are both called the moment of inertia, and confusing them is the most expensive mistake in this corner of engineering. The one on this page is the area moment of inertia, also called the second moment of area. It has units of length to the fourth power, it depends only on the shape of a cross-section, and it is what governs how much a beam bends or a column buckles. The other is the mass moment of inertia, measured in kilogram metres squared, which governs how hard a body is to spin up. They share a name, a symbol and nothing else.
The area moment of inertia measures how far a cross-section's material sits from a chosen axis, weighted by the square of that distance. Because the weighting is quadratic, geometry dominates: doubling the depth of a rectangular beam multiplies its bending stiffness by eight, while doubling its width only doubles it. That single fact explains why joists stand on edge, why an I-beam puts almost all of its steel in the flanges, and why a scaffold pole is a hollow tube rather than a solid bar of the same weight.
This moment of inertia calculator from Arb Digital returns the value about the centroidal axes of five standard shapes, and then applies the parallel-axis theorem to transfer that value to any other parallel axis you nominate. It shows the transfer term as its own number rather than folding it silently into the total, because seeing how much of the answer comes from position rather than shape is usually the point. It is a preliminary sizing and teaching tool, not a structural design: it produces geometry, never a verdict on whether a member is adequate. That decision belongs to a licensed engineer working to the governing code.
What This Moment of Inertia Calculator Does
Five sections are supported: a solid rectangle, a solid circle, a hollow circle, a hollow rectangle and a triangle. For each one the tool computes the area, the centroidal moment of inertia about both the horizontal and the vertical axis, and then the value about your offset axis. The two centroidal figures are given together deliberately, because the strong axis and the weak axis of the same section can differ by an order of magnitude, and a member rotated ninety degrees during installation is a real and common failure mode.
The offset input is what separates this page from a lookup table. Tables list centroidal values, and those are only directly useful when the axis you care about happens to pass through the centroid. In a built-up section it never does. A flange plate sitting near the top of a fabricated girder has a tiny moment of inertia about its own centroid and an enormous one about the girder's neutral axis, and the difference between them is entirely the transfer term.
The radius of gyration is reported about your chosen axis rather than the centroid, because that is the form buckling calculations need. It is the distance from the axis at which the whole area could be concentrated without changing the moment of inertia, and it is the geometric half of a slenderness ratio.
How to Use It
- Choose the shape and the unit first. The dimension labels change to match the shape, and every result is expressed in the length unit you selected raised to the fourth power.
- Enter the outside dimensions. For rectangles the width is horizontal and the height vertical, which fixes which axis is strong. For circles and tubes, enter diameters, not radii.
- Add the inner dimensions for hollow sections. They must be smaller than the outer ones, and for a box the wall does not have to be the same thickness on all four sides as long as the void is centred.
- Set the offset to the axis you actually care about. Leave it at zero for the centroidal value; enter the perpendicular distance from the shape's own centroid to your axis for anything else.
- Read the split, not just the total. The bars show how much of the answer is the shape's own stiffness and how much is position. A part that is almost all transfer term is doing its job through leverage, not through section.
The Formulas and a Worked Example
For a rectangle of width b and height h, the centroidal values are Iₓ = bh³/12 and Iₖ = hb³/12. A solid circle of diameter D gives πD⁴/64 about any centroidal axis. A tube of outside diameter D and inside diameter d gives π(D⁴ − d⁴)/64. A box is the outer rectangle minus the inner one, and a triangle of base b and height h has Iₓ = bh³/36 about the centroidal axis parallel to its base, which sits one third of the height above it.
The parallel-axis theorem then states I = Ī + Ad², where Ī is the centroidal value, A the area and d the perpendicular distance between the two parallel axes. Take the default: a rectangle 50 wide and 100 deep, offset 75 from its own centroid. The centroidal value is 50 × 100³ / 12 = 4,166,666.67 mm⁴. The area is 5,000 mm² and the transfer term is 5,000 × 75² = 28,125,000 mm⁴, so the total about the offset axis is 32,291,666.67 mm⁴. Almost eighty-seven per cent of that comes from position rather than from the plate itself — which is the whole argument for putting material in flanges. The Engineering Statics chapter on moments of inertia derives each of these integrals from first principles.
Building a Composite Section One Part at a Time
Real sections are rarely one of the five shapes on this page. A tee, a channel, an angle, a plate girder or a timber box beam is a collection of rectangles, and the standard method is to handle them one at a time and add the results. Because moments of inertia only add when they are taken about the same axis, the parallel-axis theorem is not an optional refinement in that process; it is the step that makes the addition legal at all.
The procedure has three stages. First find the centroid of the whole assembly, by taking the sum of each part's area times its own centroid position and dividing by the total area. Second, run this calculator once per part, entering that part's dimensions and setting the offset to the distance from the part's centroid to the assembly's centroid. Third, add the totals. Voids are handled by treating them as parts with negative area, so a hole is subtracted rather than added, and the same transfer term is subtracted with it.
Two errors dominate this workflow. The first is adding centroidal values directly without transferring them, which always underestimates the answer, sometimes by a factor of five or more. The second is measuring the offset from the wrong reference — from the bottom of the section rather than from the assembly centroid, for instance. The LibreTexts section on composite shapes works several full examples of the method.
Why a Tube Beats a Bar of the Same Weight
Compare a solid bar 60 mm in diameter with a tube of 60 mm outside and 50 mm inside diameter. The bar has an area of 2,827 mm² and a moment of inertia of 636,173 mm⁴. The tube has an area of 864 mm² — less than a third — and a moment of inertia of 329,376 mm⁴, which is over half. Per unit of material, the tube is roughly one and a half times as stiff in bending, because the metal it kept is the metal that was furthest from the axis and the metal it removed was doing almost nothing.
The limit on that logic is local buckling. As the wall gets thinner relative to the diameter, the tube stops failing by the whole member bending and starts failing by the wall crumpling inward, which no moment of inertia predicts. Design codes handle this with slenderness limits on the width-to-thickness ratio of each element, and those limits are why you cannot simply keep hollowing a section out. The same reasoning applies to a deep thin web, which will buckle in shear long before its impressive moment of inertia is exhausted.
Strong Axis, Weak Axis and the Rotated Member
Every non-circular section has a strong axis and a weak axis, and the ratio between them can be brutal. A 50 by 200 timber joist has a moment of inertia of 33.3 million mm⁴ about its strong axis and 2.08 million mm⁴ about its weak axis: a factor of sixteen. Laid flat instead of standing on edge, that joist deflects sixteen times as far under the same load. The calculator reports both centroidal values side by side for exactly this reason.
A circular section, solid or hollow, is the one case with no weak axis at all, which is why shafts and columns that may be loaded from any direction are usually round. Note also that the two centroidal values quoted here are principal values only for sections with an axis of symmetry. An unsymmetric shape such as an angle has principal axes that are rotated away from the obvious horizontal and vertical ones, and it will deflect out of the plane of loading. That case needs the product of inertia as well, which this page does not compute.
How This Differs From the Adjacent Arb Digital Tools
This page is the parallel-axis page: it exists to move a moment of inertia from a shape's own centroid to a different parallel axis, which is the step composite sections need. The section modulus calculator reports centroidal section properties for standard shapes and explicitly does not apply the parallel-axis theorem, so the two do not overlap. The mass moment of inertia calculator handles the rotational-dynamics quantity in kilogram metres squared, a different physical property with different units. The polar moment of inertia calculator covers torsion about the longitudinal axis rather than bending about a transverse one. The beam deflection calculator and the column buckling calculator both consume the number this page produces. As with every code-bound tool on the site, and following the precedent set by the live breaker size calculator, no allowable stress or material design value is published here.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Confusing area and mass moment of inertia — one is in length to the fourth power and governs bending, the other is in kilogram metres squared and governs spinning. A dimension check catches this instantly.
- Adding centroidal values from a table without transferring them — moments of inertia only add about a shared axis. Skipping the transfer term always understates the result, usually badly.
- Entering a radius where a diameter is asked for — the fourth power means a factor of two in the input becomes a factor of sixteen in the output.
- Mixing units within one calculation — millimetres for the section and metres for the offset produces a number that looks plausible and is wrong by twelve orders of magnitude.
- Assuming a large moment of inertia means a safe member — stiffness is not strength. Bending stress, shear, local buckling and lateral torsional buckling are separate checks, and any one of them can govern.
Related Free Tools From Arb Digital
Feed this number into the beam deflection calculator or the column buckling calculator, or compare it with the centroidal properties from the section modulus calculator. For rotation rather than bending, use the mass moment of inertia calculator, and for torsion the polar moment of inertia calculator. Stress states are handled by the stress and strain calculator and the Mohr's circle calculator, and plain geometry by the cross sectional area calculator. The full free online tools hub lists everything Arb Digital has published.
Frequently Asked Questions
A geometric property of a cross-section equal to the integral of the squared distance from an axis over the whole area. It has units of length to the fourth power and it measures how effectively a shape resists bending about that axis.
Completely, apart from the name. The area version depends only on a cross-section's geometry, is measured in units like millimetres to the fourth, and governs bending and buckling. The mass version depends on how mass is distributed in a solid body, is measured in kilogram metres squared, and governs angular acceleration.
It states that the moment of inertia about any axis equals the moment of inertia about a parallel axis through the centroid plus the area times the square of the distance between the two axes. It is what allows the properties of separate parts to be combined about a common axis.
Because the transfer term is proportional to the square of the distance. Doubling the offset quadruples the contribution, so material placed well away from the neutral axis dominates the total even when the piece itself is small.
Not in one step, because those shapes are not in the list. Split them into rectangles, find the centroid of the whole, then run the calculator once per rectangle with the offset measured from the assembly centroid, and add the totals.
Treat it as a part with negative area. Compute its own moment of inertia and its transfer term exactly as for a solid part, then subtract both from the running total instead of adding them.
It means a stiffer one. Deflection falls as the moment of inertia rises, but bending strength depends on the section modulus, and shear, local buckling and lateral stability are separate checks that a single geometric number cannot answer.
This tool is provided for educational and preliminary sizing use. It computes section geometry only. It is not a structural design, is not stamped, and does not replace a licensed engineer working to the governing building code, whose local amendments differ by jurisdiction.
