What Is the Area Moment of Inertia?
The area moment of inertia (also called the second moment of area or second moment of inertia) is a geometrical property of any cross-section. It describes how the area is distributed about an arbitrary axis, and directly determines a beam's resistance to bending and deflection. The SI unit is m⁴ (metres to the fourth power); in US-customary engineering you will encounter in⁴ or ft⁴.
We distinguish between the moment of inertia about the horizontal centroidal x-axis (denoted Ix) and about the vertical centroidal y-axis (denoted Iy). All formulas on this page assume the origin of the coordinate system coincides with the centroid of the shape.
Moment of Inertia Formulas
The following closed-form expressions give the second moment of area about the centroidal axes.
Rectangle
Ix = b · h³ / 12
Iy = h · b³ / 12
where b is the width (along x) and h is the height (along y).
Triangle
Ix = b · h³ / 36
Iy = (h · b³ − h · a · b² + b · h · a²) / 36
where b is the base, h is the height, and a is the horizontal displacement of the apex from the left end of the base (a = 0 for a right triangle, a = b/2 for an isosceles triangle).
Circle
Ix = Iy = π / 4 · r⁴
Semicircle (about the centroidal x-axis)
Ix = (π/8 − 8/(9π)) · r⁴ ≈ 0.1098 · r⁴
Iy = π/8 · r⁴
The centroid of a semicircle lies at 4r/(3π) above the diameter.
Ellipse
Ix = π/4 · a · b³
Iy = π/4 · b · a³
where a is the semi-axis along x and b is the semi-axis along y.
Regular Hexagon
Ix = Iy = 5√3 / 16 · s⁴ ≈ 0.5413 · s⁴
where s is the side length. A regular hexagon has equal moments of inertia about both centroidal axes.
How to Calculate the Moment of Inertia About Any Axis — Parallel Axis Theorem
The formulas above apply only when the axis passes through the centroid. To find the second moment of area about a parallel axis at distance d from the centroidal axis, use the Parallel Axis Theorem:
Iparallel = Icentroid + A · d²
where A is the cross-sectional area and d is the perpendicular distance between the two axes. Enter a value in the "Parallel Axis Distance" field to have the calculator apply this automatically.
Example: Moment of Inertia of a Rectangle
Suppose you have a rectangular cross-section 200 mm wide and 100 mm tall.
- Ix = 200 × 100³ / 12 = 16,666,667 mm⁴ ≈ 1.667 × 10⁷ mm⁴
- Iy = 100 × 200³ / 12 = 66,666,667 mm⁴ ≈ 6.667 × 10⁷ mm⁴
The rectangle resists bending about the y-axis four times better than about the x-axis when oriented with its longer side horizontal — which is why I-beams are oriented with the web vertical.
US Customary vs. Metric Units
The calculator supports both metric (mm, cm, m) and US customary / Imperial (in, ft) inputs. The resulting unit of the moment of inertia is always the fourth power of the chosen length unit (mm⁴, cm⁴, m⁴, in⁴, or ft⁴). In structural engineering the most common practical units are cm⁴ (metric) and in⁴ (US).
Applications of the Area Moment of Inertia
- Beam bending: The bending stress σ = M · y / I and the deflection δ ∝ 1 / (E·I) both depend on I. A larger second moment means less deflection and lower stress.
- Structural profiles: Standard I-beams and H-sections are optimised to maximise Ix with minimum material, keeping weight low.
- Column buckling: The Euler critical load Pcr = π² E I / L² uses the minimum second moment of area.
- Torsion: For thin-walled open sections, the torsional stiffness is related to the second moment of the wall thickness distribution.
- Machine design: Shafts, cranks, and levers are sized using the area moment of inertia to ensure deflection and stress remain within allowable limits.
FAQs
What is the difference between mass moment of inertia and area moment of inertia?
The area moment of inertia (second moment of area) describes a cross-section's geometry and governs bending stiffness. Its unit is length⁴ (e.g., m⁴). The mass moment of inertia describes a body's resistance to angular acceleration and has units of kg·m² (or lb·ft²). They are conceptually similar but used in different fields.
Why does a larger second moment of area mean less deflection?
The deflection of a simply supported beam under a central load is δ = F·L³ / (48·E·I). Because I appears in the denominator, doubling I halves the deflection. Deeper beams deflect less precisely because their cross-section has a larger Ix.
Can the second moment of area be negative?
No. By definition I = ∫ y² dA, and y² ≥ 0 everywhere, so the area moment of inertia is always non-negative.
What is the polar moment of inertia?
The polar moment of inertia J = Ix + Iy describes a cross-section's resistance to torsional (twisting) loads. For a solid circle, J = π/2 · r⁴.