About the Area Moment of Inertia Calculator
This area moment of inertia calculator works out the geometric properties that govern how a beam section resists bending. Choose a shape — solid or hollow rectangle, solid or hollow circle, I-beam, T-section, channel or angle — and enter its dimensions in millimetres or inches. It returns the second moment of area about both centroidal axes (Ix and Iy), the elastic section moduli (Sx and Sy), the radii of gyration, the cross-section area and the location of the centroid.
Structural and mechanical engineers, fabricators and students use these numbers in beam deflection (δ ∝ 1/EI), bending stress (σ = M/S) and column buckling (which depends on the radius of gyration) calculations. Comparing shapes quickly shows why deep I-beams and tubes are so efficient: material far from the neutral axis adds stiffness rapidly.
Sections are modelled with sharp corners and constant thickness, so rolled steel shapes with root fillets and tapered flanges will have slightly higher published values (often 2–5%). For an angle, Ix and Iy are about centroidal axes parallel to the legs, not the principal axes. The x-axis is horizontal, so Ix is the strong-axis value for an upright I-beam.
How to use the area moment of inertia calculator
- 1Choose the section shape and your length unit.
- 2Enter the overall width and height (or outside diameter for round sections).
- 3Enter the flange or wall thickness and, for I, T and channel sections, the web thickness.
- 4Read Ix, Iy, the section moduli and radii of gyration.
- 5Use Ix with Young’s modulus in deflection formulas and Sx for bending stress (σ = M ÷ S).
Formula and method
Every shape except the circles is split into rectangles (holes count as negative area). The centroid is the area-weighted average position of the pieces, and the moment of inertia about the centroid uses the parallel-axis theorem: each rectangle contributes its own b·h³/12 plus its area times the square of its distance from the overall centroid.
The elastic section modulus is the moment of inertia divided by c, the distance from the centroid to the extreme fibre — for unsymmetric sections like a T or channel, the farther side governs, giving the higher bending stress. The radius of gyration √(I/A) is used in column buckling. Circular sections use the closed-form π(D⁴ − d⁴)/64 where d is the inside diameter (zero for a solid bar).
- I
- Second moment of area (length⁴)
- b, h
- Width and height of a rectangle
- D, d
- Outside and inside diameter
- A
- Cross-section area
- dᵢ
- Distance from a piece’s centroid to the section centroid
- S
- Elastic section modulus (length³)
- c
- Distance from centroid to extreme fibre
- r
- Radius of gyration
Worked examples
I-beam 200 × 100 mm, 8.5 mm flanges, 5.6 mm web
The two flanges (100 × 8.5 mm) sit 95.75 mm from the centroid and dominate: 2 × (100 × 8.5³/12 + 850 × 95.75²) ≈ 15.59 × 10⁶ mm⁴. The 183 mm web adds 5.6 × 183³/12 ≈ 2.86 × 10⁶ mm⁴, for Ix ≈ 18.46 × 10⁶ mm⁴. Dividing by 100 mm gives Sx ≈ 184,600 mm³. (A rolled IPE 200 lists 19.4 × 10⁶ because of its root fillets.)
Solid rectangle 2 × 6 in (on edge)
Ix = 2 × 6³ ÷ 12 = 36 in⁴ and Iy = 6 × 2³ ÷ 12 = 4 in⁴, so the section is nine times stiffer on edge than flat. Sx = 36 ÷ 3 = 12 in³ and rx = √(36 ÷ 12) ≈ 1.73 in.
Pipe 60 mm OD × 4 mm wall
The inside diameter is 52 mm, so I = π × (60⁴ − 52⁴) ÷ 64 ≈ 277,264 mm⁴, the same about every axis. S = I ÷ 30 ≈ 9,242 mm³ and the polar moment J = 2I ≈ 554,529 mm⁴ for torsion.
T-section 150 wide × 120 deep, 12 mm flange, 10 mm web
The flange (1,800 mm² at y = 114) and web (1,080 mm² at y = 54) put the centroid (1,800 × 114 + 1,080 × 54) ÷ 2,880 = 91.5 mm above the bottom. Using the parallel-axis theorem, Ix ≈ 3.501 × 10⁶ mm⁴. The bottom of the web is the farthest fibre at 91.5 mm, giving Sx ≈ 38,266 mm³.
Frequently asked questions
What is the area moment of inertia?+
Also called the second moment of area, it measures how a cross-section’s area is distributed about an axis. The farther material sits from the neutral axis, the larger the value and the stiffer the beam in bending. It has units of length to the fourth power (mm⁴ or in⁴).
What is the difference between moment of inertia and section modulus?+
Moment of inertia I governs stiffness and deflection. Section modulus S = I ÷ c, where c is the distance to the outermost fibre, governs bending stress: σ = M ÷ S. A beam is chosen so that S is large enough for strength and I large enough to limit deflection.
What is the moment of inertia of a rectangle?+
About its centroidal axis parallel to the width, I = b·h³ ÷ 12. Because height is cubed, doubling the depth of a beam makes it eight times stiffer, which is why joists and beams are installed on edge.
Is area moment of inertia the same as mass moment of inertia?+
No. Area moment of inertia (length⁴) describes a cross-section’s resistance to bending. Mass moment of inertia (mass × length²) describes a body’s resistance to angular acceleration when it rotates.
Why do published steel section values differ slightly?+
Rolled steel shapes have root fillets where the web meets the flange, and some have tapered flanges. These add area near the neutral axis and corners, so published Ix values are usually a few percent higher than this sharp-corner model.