Science & Engineering
Moment of inertia of a section.
Second moment of area, section modulus and area for a rectangle, circle, hollow rectangle, tube, I-section or triangle. This is the area moment used in beam bending, not the mass moment of a spinning part.
Shape and size
Result
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Worked example
A rectangular bar 50 mm wide and 100 mm deep: Ix = b h³ ÷ 12 = 50 × 100³ ÷ 12 = 4,166,667 mm⁴ (about 4.17 × 10⁶ mm⁴), bending about the horizontal axis. Iy = h b³ ÷ 12 = 1,041,667 mm⁴, area = 5,000 mm² and section modulus Sx = Ix ÷ c = 4,166,667 ÷ 50 = 83,333 mm³. For an I-section with 100 mm flanges, 200 mm overall depth, 10 mm flanges and a 6 mm web: Ix = (100 × 200³ − 94 × 180³) ÷ 12 = 20,982,667 mm⁴.
How it works
The second moment of area (area moment of inertia) measures how a cross-section's area is spread away from its bending axis; it is the I in σ = Mc ÷ I and δ = PL³ ÷ 48EI. For a rectangle Ix = b h³ ÷ 12, for a solid circle I = π d⁴ ÷ 64, for a hollow circle I = π (D⁴ − d⁴) ÷ 64, for a hollow rectangle Ix = (B H³ − b h³) ÷ 12 and for a triangle about its centroid Ix = b h³ ÷ 36. The I-section is a wide rectangle minus the two side voids beside the web: Ix = (bf h³ − (bf − tw)(h − 2 tf)³) ÷ 12 and Iy = (2 tf bf³ + (h − 2 tf) tw³) ÷ 12; that I-section result is derived from the rectangle formula, not copied from a handbook table. Section modulus is S = I ÷ c.
For the triangle, c is the distance from the centroid to the far apex (2h ÷ 3), which gives the smaller section modulus. This is the area moment about a cross-section axis, not the mass moment of inertia used for rotating parts, which has different units (kg·m²).
Not for structural design
Planning and learning aid only. It is not for structural, pressure-system or other safety-critical design, and it is not professional engineering advice. Have a qualified engineer check anything that carries people or could fail dangerously. Standard rolled steel and timber sections have published property tables; use those for real members.
Sources and basis: Fundamental Mechanics of Materials Equations (Missouri S&T, J. Thomas); List of second moments of area (Wikipedia). Checked 2026-10-08. Planning aid only; see the note above.
FAQ
What is the moment of inertia of a rectangle?
About the centroidal axis parallel to the width b it is b × h³ ÷ 12, where h is the height perpendicular to that axis. A 50 × 100 mm bar has Ix = 4,166,667 mm⁴ about its horizontal axis.
What is the moment of inertia of a circle?
For a solid circle I = π × d⁴ ÷ 64 about any diameter. A 50 mm bar gives about 306,796 mm⁴.
Is this the same as mass moment of inertia?
No. This is the second moment of area (units length⁴) used for bending. The mass moment of inertia used for rotating bodies has units of kg·m².
What is section modulus?
S = I ÷ c, where c is the distance from the neutral axis to the outermost fibre. Bending stress is M ÷ S.
How is the I-section calculated?
As the full rectangle (flange width × overall depth) minus the two rectangular voids either side of the web. Fillets and tapered flanges on rolled sections are ignored.