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Solid rectangle — Geometric properties

Calculate area, centroid, second moments of area, section moduli and radii of gyration with clearly defined dimensions.

Created v1.0

By Thibaut Grzelak, Mechanical Analysis Engineer

8 profiles available
01

Choose a shape

02

Calculation inputs

Solid rectangle
mm
mm

Solid homogeneous section. Centroidal axes parallel to the sides. Fillets and chamfers ignored.

Results are up to date.

03

Diagram

BH
04

Results

Area A
20,000mm²
Centroid x̄
50mm
Centroid ȳ
100mm
Second moment Iₓ
66.6667E6mm⁴
Second moment Iᵧ
16.6667E6mm⁴
05Show additional results
Section modulus Wₓ (Sₓ)666,667 mm³
Section modulus Wᵧ (Sᵧ)333,333 mm³
Radius of gyration rₓ57.735 mm
Radius of gyration rᵧ28.8675 mm
Product of inertia Iₓᵧ0 mm⁴
Principal moment I₁ (max.)66.6667E6 mm⁴
Principal moment I₂ (min.)16.6667E6 mm⁴
Principal-axis angle θₚ0 °
Torsion constant J45.7363E6 mm⁴
Plastic modulus Wpl,x1E6 mm³
Plastic modulus Wpl,y500,000 mm³
Shear area Av,xNot available
Shear area Av,yNot available

Wpl is integrated exactly over the geometry. J is shown only where an exact implemented solution applies; code-dependent shear areas remain unavailable.

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Calculation note

Full calculation details

Show additional results

Wpl,x
∫A |y − yp| dA
Wpl,y
∫A |x − xp| dA
J
J = ab³/3 · [1 − 192b/(π⁵a) · Σₙ₌₁,₃,… tanh(nπa/2b)/n⁵]
Av,x / Av,y
Not available

Wpl is integrated exactly over the geometry. J is shown only where an exact implemented solution applies; code-dependent shear areas remain unavailable.

01Equations and symbols

Formulas used

Ab · h
x̄b / 2
ȳh / 2
Iₓb · h³ / 12
Iᵧh · b³ / 12
WₓIₓ / (h / 2)
WᵧIᵧ / (b / 2)
rₓ√(Iₓ / A)
rᵧ√(Iᵧ / A)
SymbolMeaningDimension
bWidth bL
hHeight hL
ASection areaL²
x̄Centroid x-coordinateL
ȳCentroid y-coordinateL
IₓSecond moment about xL⁴
IᵧSecond moment about yL⁴
WₓSection modulus about xL³
WᵧSection modulus about yL³
rₓRadius of gyration about xL
rᵧRadius of gyration about yL
02Assumptions and limits

Assumptions and limits

  • Solid homogeneous section.
  • Centroidal axes parallel to the sides.
  • Fillets and chamfers ignored.

This calculation does not check strength, stability or code compliance.

03Calculator validation

Calculator validation

The engine is checked against analytical identities and known scaling laws.

  • 100 × 200 mm case: A = 20,000 mm², Iₓ = 66,666,666.7 mm⁴.
  • A square satisfies Iₓ = Iᵧ.
  • Doubling dimensions gives A × 4 and I × 16.
  • Non-positive dimensions are rejected.
04References

References

  1. Roark’s Formulas for Stress and Strain, 9th edition, section-property tables.
  2. Young, Budynas & Sadegh, Roark’s Formulas for Stress and Strain, McGraw-Hill.
  3. BIPM, The International System of Units (SI), 9th edition.

Calculation path and sources

Section properties can be reused for linear mass, bending and buckling checks.

Reuse the results

Continue with a mechanical calculation

Geometric properties become inputs for mass, stress, deformation and stability checks.

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External technical sources

Geometric properties

Second moment of area and section modulus: which value to use?

I describes material distribution and W connects geometry to bending stress. Values change with the x or y axis.

Second moment of area I

Also called area moment of inertia, I is used in deflection and buckling.

I [mm⁴, cm⁴, m⁴, in⁴]

Elastic section modulus W

W uses distance c to the extreme fibre. For an unsymmetrical section, check both sides.

W = I / c

Choose the correct shape

Rectangle, circle, tube, I, channel, T and angle sections require different geometries.

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