What is the second moment of area formula for a rectangle?

I = b·h³/12 about the centroidal axis parallel to the width b. For bending about the other axis it is h·b³/12. The depth, not the width, is the cubed term — which is why a deeper section is far stiffer than a wider one of the same area.

How does section modulus relate to moment of inertia?

S = I/c, where c is the distance from the neutral axis to the extreme fibre — half the depth for a symmetric section. I governs deflection and has units of in⁴; S governs stress and has units of in³. A member can be adequate in one and not the other, so both are checked in design.

What is the radius of gyration used for?

Column buckling. The slenderness ratio K·L/r combines the effective length with the radius of gyration, and a column with a larger r resists buckling better for the same area. r = √(I/A), so it is the distance at which the whole area could be concentrated to give the same I.

Why do my calculated values differ from the steel manual?

Fillets. Real rolled shapes have a rounded transition between the web and the flange that the idealised formula ignores, adding 2–3% to the moment of inertia and the area. The published table values include them and are the ones to design from; the formulas here are for checking and estimating.

What is the difference between the elastic and plastic section modulus?

The elastic modulus S assumes the section is still elastic, with stress varying linearly from zero at the neutral axis. The plastic modulus Z describes the fully yielded state, where the whole section carries the yield stress. Z is about 10–15% larger than S for an I-section and is used in plastic design methods.

Section Properties

Area, second moment of area, elastic section modulus and radius of gyration for nine common cross-sections, with the axis each value refers to stated.

Data verified 2026-09-29 · based on AISC 15th edition (2017), Shapes Database v15.0

Quick Answer

Ix = b·h³ ÷ 12 for a rectangle about its centroid, and Sx = b·h² ÷ 6. Depth appears cubed in I and squared in S, which is why adding depth is the cheapest way to stiffen a section. The centroidal axis is the one that matters — using a base axis instead gives three to four times the value.

Find a shape

Section Properties by Shape

Cross-section^[2]	Axis the values refer to^[2]	Area, A^[1]	Second moment of area, I^[1]	Elastic section modulus, S^[1]	Radius of gyration, r^[1]
Rectangle, b × h #	Bending about the centroidal axis parallel to b	A = b·h	b·h³ ÷ 12	b·h² ÷ 6	h ÷ √12 = 0.2887h
Square, a × a #	Any axis through the centroid	A = a²	a⁴ ÷ 12	a³ ÷ 6	a ÷ √12 = 0.2887a
Solid circle, diameter d #	Any axis through the centre	A = π·d² ÷ 4	π·d⁴ ÷ 64	π·d³ ÷ 32	d ÷ 4
Hollow circle, od × id #	Any axis through the centre	A = π(od² − id²) ÷ 4	π(od⁴ − id⁴) ÷ 64	π(od⁴ − id⁴) ÷ (32·od)	√(od² + id²) ÷ 4
Hollow rectangle, b × h × t #	Uniform wall thickness t, bending about the parallel axis	A = b·h − (b−2t)(h−2t)	[b·h³ − (b−2t)(h−2t)³] ÷ 12	[b·h³ − (b−2t)(h−2t)³] ÷ (6h)	√(Ix ÷ A)
I-section, idealised #	No fillets: d, flange width bf, web tw, flange tf	A = 2·bf·tf + (d−2tf)·tw	[bf·d³ − (bf−tw)(d−2tf)³] ÷ 12	2·Ix ÷ d	√(Ix ÷ A)
Triangle, base b, height h #	About the centroidal axis, parallel to the base	A = b·h ÷ 2	b·h³ ÷ 36	b·h² ÷ 24	h ÷ √18 = 0.2357h
Thin-walled tube, mean radius r, thickness t #	Wall thickness much less than radius	A = 2π·r·t	π·r³·t	π·r²·t	r ÷ √2 = 0.7071r
Rectangle, about its base #	Not a centroidal axis — use for cantilever checks	A = b·h	b·h³ ÷ 3	b·h² ÷ 3	h ÷ √3 = 0.5774h

Values are for the centroidal axis unless the row says otherwise — the axis through the centre of area. This matters more than anything else on the page: a rectangle about its base has I = b·h³/3, four times the centroidal value, and a beam and a cantilever are checked about different axes. S is the elastic section modulus, used for design against yield; the plastic modulus Z is 10–15% larger for an I-section and is used in plastic design. Values are for the stated shape only — holes, notches, tapers and the fillets in a real rolled section are not represented.

Which Axis Are You Checking?

Every section has two principal moments of inertia, one for bending about each axis, and which one governs depends on how the member is loaded and supported.

A floor joist spanning between two walls bends about its strong axis and is checked for Ix. The same joist braced laterally against buckling is checked for Iy about the weak axis. A cantilever is also checked about the same axes but with the base condition, where I is four times the centroidal value because the neutral axis is not at mid-depth.

Getting the axis wrong by a factor of four is the classic error, and it is invisible in a calculation that otherwise looks right. State the axis on the drawing.

Rolled Sections Need the Published Table

An I-beam or channel can be computed as the difference of rectangles using the idealised formula in the table, but the answer will be 2 to 3% low — real rolled shapes have fillets between the web and the flange, which add material exactly where it contributes most to stiffness.

That is why design values come from a published section property table rather than from geometry. Use the idealised formula here for a sanity check or an estimate, and take Ix, Sx, rx and Zx from the AISC Shapes Database or the equivalent national table for anything that will be built.

Composite and built-up sections need the parallel axis theorem on top of these formulas: I about a new axis = I_centroid + A·d², where d is the distance the axis moved. T-sections and unsymmetrical built-up beams cannot be handled by the simple formulas above.

Frequently Asked Questions

What is the second moment of area formula for a rectangle?: I = b·h³/12 about the centroidal axis parallel to the width b. For bending about the other axis it is h·b³/12. The depth, not the width, is the cubed term — which is why a deeper section is far stiffer than a wider one of the same area.
How does section modulus relate to moment of inertia?: S = I/c, where c is the distance from the neutral axis to the extreme fibre — half the depth for a symmetric section. I governs deflection and has units of in⁴; S governs stress and has units of in³. A member can be adequate in one and not the other, so both are checked in design.
What is the radius of gyration used for?: Column buckling. The slenderness ratio K·L/r combines the effective length with the radius of gyration, and a column with a larger r resists buckling better for the same area. r = √(I/A), so it is the distance at which the whole area could be concentrated to give the same I.
Why do my calculated values differ from the steel manual?: Fillets. Real rolled shapes have a rounded transition between the web and the flange that the idealised formula ignores, adding 2–3% to the moment of inertia and the area. The published table values include them and are the ones to design from; the formulas here are for checking and estimating.
What is the difference between the elastic and plastic section modulus?: The elastic modulus S assumes the section is still elastic, with stress varying linearly from zero at the neutral axis. The plastic modulus Z describes the fully yielded state, where the whole section carries the yield stress. Z is about 10–15% larger than S for an I-section and is used in plastic design methods.

Value Sources

Each data column on this page is tied to the source it came from. The numbers in square brackets correspond to the table headers above.

#	Source	Type	Revision / method
[1]	Value computed from the standard's defining relationship	derived	Computed at build time from the defining formula stated on the page, then verified against every row and anchored by known standard values.
[2]	AISC Steel Construction Manual, Shapes Database	standard	AISC 15th edition (2017), Shapes Database v15.0

Data Sources

Standard	Revision	What it covers on this page
AISC Steel Construction Manual, Shapes Database	AISC 15th edition (2017), Shapes Database v15.0	the shape definitions and the axes the properties refer to
ASTM A6/A6M — General Requirements for Rolled Structural Steel Bars, Plates, Shapes and Sheet Piling	ASTM A6/A6M-24	the nominal dimensions of rolled shapes
Roark's Formulas for Stress and Strain	Roark's, 8th edition (2011)	the closed-form section property formulas

Cross-checked against:

The idealised I-section formula checked against published AISC section property values, confirming the expected 2–3% fillet difference
The rectangle and circle formulas checked against their known closed forms at several sizes
The radius of gyration relationships verified as r = √(I/A) for every row

Derived values — the following values on this page are calculated, not taken directly from the standard:

Value	How it is derived
Area, I, S and r	Standard closed-form formulas for each shape, evaluated in the browser by the linked calculators and stated symbolically here. The radius of gyration entries are restatements of r = √(I/A) rather than independent values.

Values refer to the stated axes, normally centroidal. Rolled sections need published table values because fillets add 2–3% that these idealised formulas omit. Holes, notches, tapers, composite sections and unsymmetrical built-up members are not covered — those need the parallel axis theorem or a published table.

Accuracy and use. The values on this page are compiled from the published standards and cross-checked sources listed above. Where values are derived, the derivation is stated. No warranty, express or implied, is made as to the accuracy or completeness of this information, and no liability is accepted for any loss or damage arising from its use. Engineering reference data is provided for guidance in preliminary work — before a value is used for design, fabrication or acceptance testing, verify it against the current revision of the governing standard and against your own inspection. The user assumes all risk and responsibility in connection with the use of this information.

Cite This Page

Every value on this page is traceable to the sources listed above. If you use the data in a document, paper or report, cite it as:

Plain text

Engineering Reference. "Section Properties — Moment of Inertia & Section Modulus." https://refmanual.com/charts/section-properties/ (accessed 2026-09-29). Data from AISC Steel Construction Manual, Shapes Database, AISC 15th edition (2017), Shapes Database v15.0.

BibTeX

@misc{section-properties,
  title        = {Section Properties — Moment of Inertia & Section Modulus},
  author       = {{Engineering Reference}},
  howpublished = {\url{https://refmanual.com/charts/section-properties/}},
  note         = {Accessed: 2026-09-29},
  source       = {AISC Steel Construction Manual, Shapes Database, AISC 15th edition (2017), Shapes Database v15.0}
}

RIS (EndNote, Zotero, Mendeley)

TY  - ELEC
TI  - Section Properties — Moment of Inertia & Section Modulus
AU  - Engineering Reference
PB  - Engineering Reference
UR  - https://refmanual.com/charts/section-properties/
PY  - 2026
DA  - 2026-09-29
N1  - Source: AISC Steel Construction Manual, Shapes Database, AISC 15th edition (2017), Shapes Database v15.0
ER  -

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