Best for
Beam stiffness checks
Use I with E and span/loading formulas to compare deflection and bending stiffness.
Engineering formula reference
Second Moment of Area Formula Sheet
Search second moment of area formulas for common sections, with variables explained for beam bending, stiffness, deflection, and section comparison.
Section properties
Search second moment of area formula sheet
Filter the formula table by shape, variable, symbol, expression, or engineering use case.
Axis matters
Confirm the reference axis
These formulas assume the listed centroidal axis unless a row states otherwise.
Formula reference
Second moment of area formulas with variables
| Shape | Second moment of area | Variables and notes |
|---|---|---|
| Rectangle about centroidal x-axis | Ix = b h3 / 12 | b = width, h = height |
| Circle | I = π r4 / 4 | r = radius |
| Hollow circle | I = (π / 4)(R4 - r4) | R = outer radius, r = inner radius |
| Triangle about centroidal x-axis | Ix = b h3 / 36 | Axis parallel to base through centroid |
| Thin-walled circular tube | I ≈ π r3 t | Approximation for t much smaller than r |
| I-beam simplified | I ≈ b h3/12 - bihi3/12 | Outer rectangle minus inner void approximation |
| Box section | I = b h3/12 - (b - 2t)(h - 2t)3/12 | Strong-axis rectangular tube approximation |
| Square | I = a4 / 12 | a = side length |
| Regular hexagon | I = (5√3 / 16) a4 | a = side length |
How to use
Practical engineering checks
Beam deflection
δ depends on E I
E is modulus of elasticity and I is second moment of area.
Bending stress
σ = M y / I
M is bending moment and y is distance from the neutral axis.
Section modulus
S = I / c
c is distance from neutral axis to the extreme fiber.
Parallel axis theorem
I = Ic + A d2
Use this when moving an area moment from a centroidal axis.
Definitions
Symbols and variables explained
I
Second moment of area about the bending axis.
Ix
Second moment of area about the x-axis.
c
Distance from neutral axis to the outermost fiber.
E I
Flexural rigidity used in beam deflection calculations.
FAQ
Second Moment of Area Formula Sheet questions
How do I use this formula sheet?
Find the shape or identity, confirm each variable definition, keep units consistent, then substitute values into the formula.
Are these formulas exact?
The basic geometry formulas are exact for ideal shapes. Rows marked approximate depend on simplified geometry, thin-wall assumptions, or ignoring radii and fillets.
Can I use these formulas for final engineering design?
Use them as calculation references. Final design should also check loads, materials, tolerances, safety factors, and any governing standards.
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