Second Moment of Area Calculator
Instantly calculate the area moment of inertia (Ix, Iy), section modulus, and radius of gyration for rectangular, circular, hollow, I-beam, T-beam, and channel cross-sections.
Cross-Section Properties
Select a section shape and enter its dimensions to calculate I, Z, and r
Wall thickness = (B−b)/2 horizontally, (D−d)/2 vertically
Wall thickness = (D−d)/2
Total depth = hw + 2×tf. Top and bottom flanges assumed equal.
Total depth = hw + tf. Centroid calculated from bottom of web.
Section Properties
Ix, Iy, section modulus & radius of gyration
Select a cross-section shape, enter its dimensions, then click Calculate Section Properties to see Ix, Iy, section modulus, and radius of gyration.
Second Moment of Area Formulas
Standard closed-form expressions for the area moment of inertia (Ix) about the centroidal x-axis for common structural cross-sections. All formulas assume bending about the horizontal centroidal axis unless noted.
| Section | Ix (centroidal, horizontal axis) | Iy (centroidal, vertical axis) | Cross-sectional Area (A) |
|---|---|---|---|
| Solid Rectangle (b × d) | bd³ / 12 | db³ / 12 | b × d |
| Solid Circle (diameter D) | πD⁴ / 64 | πD⁴ / 64 | πD² / 4 |
| Hollow Rectangle (B×D outer, b×d inner) | (BD³ − bd³) / 12 | (DB³ − db³) / 12 | BD − bd |
| Hollow Circle (D outer, d inner) | π(D⁴ − d⁴) / 64 | π(D⁴ − d⁴) / 64 | π(D² − d²) / 4 |
| I-Beam (bf, tf, hw, tw) | bf·(hw+2tf)³/12 − (bf−tw)·hw³/12 | tw³·hw/12 + 2·tf·bf³/12 | 2·bf·tf + hw·tw |
| T-Beam (bf, tf, hw, tw) | Parallel axis theorem (see below) | tw³·hw/12 + tf·bf³/12 | bf·tf + hw·tw |
| Property | Symbol | Formula | Units |
|---|---|---|---|
| Second Moment of Area | Ix, Iy | ∫y² dA (centroidal axis) | mm⁴ / m⁴ / in⁴ |
| Elastic Section Modulus | Zx = Sx | Ix / y_max | mm³ / m³ / in³ |
| Radius of Gyration | rx | √(Ix / A) | mm / m / in |
| Polar Moment of Inertia | J | Ix + Iy | mm⁴ / m⁴ / in⁴ |
| Parallel Axis Theorem | I = Ic + Ad² | Ic: centroidal I, d: distance to new axis | mm⁴ / m⁴ / in⁴ |
Second Moment of Area FAQ
Everything you need to know about calculating and applying the area moment of inertia in structural engineering and beam design.
The second moment of area (also called area moment of inertia) is a geometric property of a cross-section that measures its resistance to bending. It is denoted I and has units of length to the fourth power (mm⁴ or m⁴). A higher second moment of area means the section is stiffer and deflects less under the same bending load. It appears directly in the Euler-Bernoulli beam equation: M/I = σ/y = E/R.
For a solid rectangle of width b and height (depth) d, the second moment of area about the horizontal centroidal x-axis is Ix = (b × d³) / 12. About the vertical centroidal y-axis it is Iy = (d × b³) / 12. Note that d appears cubed in Ix — this means doubling the depth increases the bending stiffness by a factor of eight, which is why deep beams are far more efficient than wide, shallow ones.
In structural engineering, these terms are often used interchangeably to refer to the area moment of inertia (I), a purely geometric property measured in mm⁴ or m⁴. In physics and dynamics, moment of inertia refers to the mass moment of inertia (kg·m²), which describes resistance to rotational acceleration. This calculator computes the geometric area moment of inertia used in beam bending, deflection, and buckling calculations.
The elastic section modulus Z (also written S) is defined as Z = I / y, where y is the distance from the neutral axis to the extreme fibre. It has units of mm³ or m³. The bending stress at any fibre is σ = M / Z (or σ = M·y / I), where M is the applied bending moment. A higher section modulus means the section can carry more bending moment before reaching its design stress limit. For a symmetric section, Z_top = Z_bottom = I / (d/2).
I-beams concentrate material in the flanges, which are located far from the neutral axis. Since the second moment of area is proportional to the square of the distance from the neutral axis (via the parallel axis theorem: I = Ic + A·d²), placing material far from the centre dramatically increases I. This makes I-beams extremely efficient in bending — providing high stiffness and strength while using less material than a solid rectangular section of the same depth.
The parallel axis theorem states: I = Ic + A·d², where Ic is the second moment of area about the centroidal axis, A is the cross-sectional area, and d is the perpendicular distance between the centroidal axis and the new axis. This theorem is used to calculate the second moment of area of composite sections (such as I-beams and T-beams) by summing the contributions of individual rectangles, each transferred to the overall centroidal axis.
The radius of gyration r is the distance from the axis at which all of the area could be concentrated to give the same second moment of area: r = √(I / A). It has units of length (mm or m). The radius of gyration is used in column buckling design — the slenderness ratio λ = L_eff / r, where L_eff is the effective buckling length. Slender columns (high λ) are prone to Euler buckling and have reduced load-carrying capacity.
The second moment of area has units of length to the fourth power. In structural engineering, mm⁴ is most common for section-level calculations (individual beams and columns), while m⁴ is used at the whole-structure or frame level. To convert: 1 m⁴ = 10¹² mm⁴ = 1×10¹² mm⁴. In US practice, in⁴ (inches to the fourth power) is standard. This calculator lets you select your preferred unit system and displays all results consistently.