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 can an idealised value differ from a manufacturer table?

Actual geometry, fillets and dimensional conventions may differ. There is no universal percentage correction; use the selected product or section database.

Is plastic modulus always 10–15% greater?

No. The ratio depends on the cross-section geometry and the assumed material model.

Section Properties Chart — Moment of Inertia & Modulus

Identified content errors were corrected on 2026-10-25. Sources and scope for this page: AISC Shapes Database v16.0 (accessed 2026-10-25), covering the source for published structural-steel section properties. This page’s ideal-shape formulas and example are geometric derivations, not transcriptions of rolled-section data. What the 2026-10-25 correction changed is stated with the source above; nothing beyond that scope was re-checked. Full sources, values and limits on this page.

Page last changed 2026-11-08 · based on Public reference accessed 2026-10-25; scope stated on page

Quick Answer

For a rectangle about its centroidal axis parallel to width b, I = bh³/12 and S = bh²/6. A beam’s support conditions change its moment and deflection equations, not the cross-section’s geometric properties about a specified axis.

Find a shape

Section Properties by Shape

Cross-section^[3]	Axis the values refer to^[3]	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 #	Centroidal axis parallel to a side (S as tabulated)	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 #	Centroidal axis parallel to width b; ideal sharp corners	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	Bottom: bh²/12; top: bh²/24	h ÷ √18 = 0.2357h
Thin-walled tube, mean radius r, thickness t #	Centroidal diameter; thin-wall approximation t ≪ r	A = 2π·r·t	π·r³·t	π·r²·t	r ÷ √2 = 0.7071r
Rectangle, about its edge #	Geometric reference axis parallel to b; not the elastic neutral axis	A = b·h	b·h³ ÷ 3	Not used as elastic bending S about this edge	h ÷ √3

Axes are cross-section axes. A fixed support at the end of a cantilever does not move the cross-section neutral axis to its edge. The triangle has two different extreme-fibre distances and hence two elastic section moduli. Hollow and built-up formulas use ideal geometry; thin-wall formulas are approximations.

Support condition is not a section axis

An actual 2 × 4 in rectangular section has centroidal Ix = 10.6667 in⁴ whether used in a simply supported beam or a cantilever bending about x. The support changes the loading solution. The separate geometric edge-axis value, 42.6667 in⁴, follows Iedge = Icentroid + A(h/2)² and is not a replacement I for cantilever bending.

Use actual section data

For manufactured sections use the applicable section-property table. Fillets, dimensions and product geometry can differ from the idealised formulas; no universal 2–3% correction is valid. Holes, composite materials and unsymmetrical built-up sections require their own section calculation.

Basis and scope

AISC Shapes Database v16.0: Source for published structural-steel section properties. This page’s ideal-shape formulas and example are geometric derivations, not transcriptions of rolled-section data.

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 can an idealised value differ from a manufacturer table?: Actual geometry, fillets and dimensional conventions may differ. There is no universal percentage correction; use the selected product or section database.
Is plastic modulus always 10–15% greater?: No. The ratio depends on the cross-section geometry and the assumed material model.

Value Sources

References and calculation methods used for this page are listed below. Numbered markers connect table columns where present. A reference listing does not itself establish that every value has been independently checked.

#	Source	Type	Revision / method
[1]	Geometric section-property formulas	derived	Calculated from the stated geometry and formulas on this page; not a product certification.
[2]	AISC Shapes Database v16.0	reference	Public reference accessed 2026-10-25 — source
[3]	Stated ideal section geometry	derived	Calculated from the stated geometry and formulas on this page; not a product certification.

Data Sources

Standard	Revision	What it covers on this page
AISC Shapes Database v16.0	Public reference accessed 2026-10-25; scope stated on page	Source for published structural-steel section properties. This page’s ideal-shape formulas and example are geometric derivations, not transcriptions of rolled-section data.

Verification scope

2026-10-25 — Content corrected 2026-10-25. Source for published structural-steel section properties. This page’s ideal-shape formulas and example are geometric derivations, not transcriptions of rolled-section data. Evidence

Content corrected 2026-10-25. Source for published structural-steel section properties. This page’s ideal-shape formulas and example are geometric derivations, not transcriptions of rolled-section data.

Accuracy and use. Use the source notes and recorded verification scope above to assess each value. Calculated values follow the stated derivations; compilation values may lack a confirmed original document or product condition. 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

Read the source notes, calculation methods and verification scope above before reusing a value. To cite this page:

Plain text

Engineering Reference. "Section Properties — Moment of Inertia & Section Modulus." https://refmanual.com/charts/section-properties/ (accessed 2026-11-09). Data from AISC Shapes Database v16.0, Public reference accessed 2026-10-25; scope stated on page.

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: 2026-11-09},
  source       = {AISC Shapes Database v16.0, Public reference accessed 2026-10-25; scope stated on page}
}

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  - 2026-11-09
N1  - Source: AISC Shapes Database v16.0, Public reference accessed 2026-10-25; scope stated on page
ER  -

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