Moment of Inertia Calculator (Common Shapes)
Moment of inertia calculator. Returns the second moment of area, section modulus and radius of gyration for rectangles, circles, tubes, box and I sections. Educational.
Moment of Inertia Calculator
Please tick the acknowledgement above before calculating.
How to use the moment of inertia calculator
Choose the section shape
Select a rectangle, solid circle, tube, box, or I-beam. The input fields change to match the dimensions that shape needs.
Enter the dimensions in mm
Type the relevant widths, heights, diameters, or thicknesses in millimetres. For symmetric sections the bending axis (x) is horizontal through the centroid.
Acknowledge the disclaimer
These are textbook formulas for ideal shapes. Tick the acknowledgement to reveal the results.
Read I, S, r and A
You get the moment of inertia (cm⁴), section modulus (cm³), radius of gyration (mm), and area (cm²) — the properties that feed deflection, bending-stress, and buckling checks.
Moment of inertia — the shape's resistance to bending
Why shape beats material for stiffness
The moment of inertia of an area — properly the second moment of area, written I — is the single number that captures how a cross-section's geometry resists bending. It appears in almost every beam and column formula: deflection is inversely proportional to I, bending stress depends on the section modulus derived from it, and buckling load scales directly with it. Crucially, I depends only on the shape and size of the cross-section, not on the material. That's why an aluminium I-beam can outperform a solid steel bar of the same weight in bending: aluminium is less stiff as a material, but arranging the metal far from the neutral axis multiplies I enough to win. The mathematics rewards distance dramatically — I integrates area times the square of its distance from the bending axis, so material at the edges counts far more than material near the middle. For a rectangle, I about the horizontal axis is b·h³/12, which means the height is cubed: doubling the depth of a beam multiplies its stiffness by eight, while doubling the width only doubles it.
This single insight explains the shape of structural sections. I-beams, channels, and hollow tubes all concentrate material away from the centre, maximising I for a given amount of steel. A solid circle has I equal to π·d⁴/64; hollow it out into a tube and you lose only the lightly stressed core, keeping almost all the bending resistance at a fraction of the weight. The section modulus, S = I divided by the distance to the extreme fibre, then converts I into a strength property: the bending stress is simply the moment divided by S. And the radius of gyration, r = √(I/A), governs column buckling. One geometric quantity, three structural consequences.
"For a rectangle, stiffness goes with the cube of depth: double the height and you octuple the resistance to bending. That's why structural sections push material to the edges — distance from the neutral axis is everything."
Reading the results correctly
This calculator returns I about the centroidal x-axis (the usual bending axis), the section modulus, the radius of gyration, the cross-sectional area, and — where the shape allows — I about the y-axis. A few cautions apply. The formulas assume idealised, sharp-cornered shapes; real rolled sections have fillets and root radii that change the values slightly, which is why engineers use published section tables for standard steel members rather than hand formulas. For an I-beam, the figure given here is for a doubly symmetric section about its strong axis using the simple subtraction method; built-up, asymmetric, or composite sections need the parallel-axis theorem applied piece by piece. And the moment of inertia is only one input to a design — it tells you about stiffness and feeds the strength and stability checks, but it does not by itself confirm a member is adequate. Use these numbers to compare sections and to feed our beam-deflection and Euler-buckling calculators, and rely on verified section data and a licensed engineer for any real design.
10 Facts About Moment of Inertia
Properly the second moment of area, symbol I.
It depends on shape only, not the material.
Rectangle: I = b·h³/12 — height is cubed.
Double the depth → 8× the stiffness.
Solid circle: I = π·d⁴/64.
Material far from the axis counts as distance squared.
Section modulus S = I/c gives bending strength.
Radius of gyration r = √(I/A) governs buckling.
Tubes keep stiffness while shedding the dead core.
Real rolled sections use published tables (fillets matter).
Frequently asked questions
It is the second moment of area, a geometric property that measures how a cross-section resists bending about a given axis. Mathematically it integrates each bit of area multiplied by the square of its distance from the axis, so material far from the centre contributes far more. It appears in deflection, bending-stress, and buckling formulas. Note it is different from the mass moment of inertia used in rotational dynamics, despite the shared name.
About the centroidal x-axis: a rectangle is b·h³/12; a solid circle is π·d⁴/64; a hollow circle (tube) is π(D⁴−d⁴)/64; a box (hollow rectangle) is (B·H³ − b·h³)/12; and a doubly symmetric I-beam is [bf·H³ − (bf−tw)(H−2tf)³]/12. The calculator applies the right formula for the shape you pick and also reports section modulus, radius of gyration, and area.
Because for bending about the horizontal axis the height appears cubed in b·h³/12, while the width appears only to the first power. Doubling a rectangle's depth multiplies its moment of inertia — and so its bending stiffness — by eight, whereas doubling its width only doubles it. This is why beams are oriented with their long dimension vertical and why a plank on edge is far stiffer than the same plank laid flat.
The section modulus S equals I divided by the distance from the neutral axis to the extreme fibre (c). While I governs deflection (stiffness), S governs bending strength: the maximum bending stress is the bending moment divided by S. Two sections can have similar I but different S if their extreme fibres are at different distances. The calculator reports both so you can do stiffness and strength checks.
The radius of gyration, r = √(I/A), expresses how far from the axis the area is effectively concentrated. It is central to column design: the slenderness ratio (effective length divided by r) determines whether a column fails by buckling or by yielding, and it feeds the Euler buckling formula. A larger radius of gyration means a more buckling-resistant column for the same area. The calculator reports it in millimetres.
Because the material near the centre of a section is close to the neutral axis and barely contributes to bending resistance. Removing it — making a tube or hollow box — sheds weight while keeping almost all the moment of inertia, which lives in the outer material. This gives a much better stiffness-to-weight ratio, which is why bicycle frames, scaffolding, and structural hollow sections are tubular. The calculator's tube and box options show exactly how much I survives.
Rolled steel sections have fillets, root radii, and rounded corners that the idealised formulas ignore, so the true properties differ slightly from a sharp-cornered calculation. Manufacturers and standards bodies publish precise section tables (I, S, r, A and more) measured for each designation. Engineers use those tables for real members. This calculator is ideal for custom or simple shapes and for understanding how geometry drives the properties.
The built-in shapes are symmetric, so the neutral axis sits at mid-height. Asymmetric sections (such as an unequal-leg angle or a tee) have a centroid that isn't at the geometric centre, and built-up or composite sections must be assembled from parts using the parallel-axis theorem. Those need a more general calculation than this tool provides. For symmetric rectangles, circles, tubes, boxes, and I-beams, the results here are exact for the idealised geometry.
Use it for learning, comparing sections, and feeding preliminary deflection or buckling estimates. The moment of inertia is only one input to a structural design, which also requires load analysis, strength and stability checks, connection design, and code compliance. Any real structure must be designed and verified by a licensed engineer using validated data. Treat these numbers as an educational aid, not a design.
No. The dimensions you enter are processed entirely in your browser. Nothing is sent to a server, stored, or shared, and no account is required. The calculation runs on your device only.
Related News
You may be interested in these recent stories from our newsroom.
No related news yet for this tool. Our editorial team publishes new pieces every week.
Browse all news →75 more free tools
Calculators, converters, security tools — no signup.