Moment of Inertia Calculator — Second Moment of Area for Structural Sections

Compute the moment of inertia for rectangles, circles, hollow sections, and I-beams instantly. Free engineering calculator with centroidal axis formulas, step-by-step breakdowns, and multiple unit options.

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Moment of Inertia Calculator

Select a cross-section shape, enter dimensions, and choose your unit system to calculate the second moment of area about the centroidal axis.

Select a shape, enter dimensions, and click Calculate Moment of Inertia to see the result.

Moment of Inertia Formulas & Equations

The moment of inertia (second moment of area) measures a cross-section's resistance to bending about a given axis. Below are the standard centroidal axis formulas for each supported shape:

ShapeFormula (Ix about centroidal axis)
Solid RectangleIx = b × h³ / 12
Solid CircleIx = π × d⁴ / 64
Hollow RectangleIx = (B × H³ − b × h³) / 12
Hollow CircleIx = π × (D⁴ − d⁴) / 64
I-BeamIx = [bf × h³ − (bf − tw) × (h − 2tf)³] / 12

Variable Definitions

  • b — Base width of rectangular section
  • h — Height of rectangular section
  • d — Diameter of circular section
  • B, H — Outer base and height of hollow rectangle
  • b, h — Inner base and height of hollow rectangle
  • D, d — Outer and inner diameters of hollow circle
  • bf — Flange width of I-beam
  • tw — Web thickness of I-beam
  • tf — Flange thickness of I-beam

How to Calculate Moment of Inertia

Follow these steps to determine the second moment of area for any structural cross-section:

  1. Identify the cross-section shape — Determine whether your section is rectangular, circular, hollow, or an I-beam profile.
  2. Measure all required dimensions — For a rectangle, measure base (b) and height (h). For an I-beam, measure flange width, total height, web thickness, and flange thickness.
  3. Select the correct formula — Use the appropriate centroidal axis formula for your shape from the table above.
  4. Substitute values and compute — Plug the measured dimensions into the formula. Remember that height terms are cubed, so small changes in depth significantly affect the result.
  5. Verify units — The moment of inertia carries units of length⁴ (e.g., mm⁴, cm⁴, in⁴). Ensure your input units are consistent.

Moment of Inertia Worked Examples

Example 1: Solid Rectangular Beam

Calculate Ix for a rectangular beam with b = 100 mm and h = 200 mm.

Ix = b × h³ / 12 = 100 × (200)³ / 12
= 100 × 8,000,000 / 12
= 66,666,667 mm⁴

Example 2: Solid Circular Rod

Calculate Ix for a solid circular rod with d = 80 mm.

Ix = π × d⁴ / 64 = π × (80)⁴ / 64
= π × 40,960,000 / 64
≈ 2,010,619 mm⁴

Example 3: Hollow Circular Pipe

Calculate Ix for a hollow pipe with D = 120 mm and d = 80 mm.

Ix = π × (D⁴ − d⁴) / 64
= π × (207,360,000 − 40,960,000) / 64
≈ 8,168,141 mm⁴

Real-World Moment of Inertia Applications

  • Structural Beam Design: Engineers use Ix to predict beam deflection and ensure structures meet serviceability requirements under dead and live loads.
  • Column Buckling Analysis: The moment of inertia determines the radius of gyration, which is critical for Euler buckling calculations in steel and timber columns.
  • Machine Shaft Design: Rotating shafts require sufficient moment of inertia to resist bending from transverse loads and maintain alignment.
  • Automotive Chassis: Vehicle frame rails are designed with specific cross-sectional moments of inertia to manage crash energy absorption and torsional stiffness.
  • Aerospace Structures: Aircraft wing spars and fuselage stringers rely on optimized moment of inertia for lightweight yet stiff structural performance.
  • Bridge Engineering: Bridge girders use I-beam and box sections with large moments of inertia to span long distances with minimal deflection.

People Also Ask

Moment of inertia, also called the second moment of area, quantifies a cross-section's resistance to bending. It depends purely on the shape's geometry and the axis about which bending occurs. A larger moment of inertia means greater stiffness and less deflection under load.
For a solid rectangular cross-section, the moment of inertia about the centroidal axis parallel to the base is I = (b × h³) / 12, where b is the base width and h is the height. About the vertical centroidal axis, use I = (h × b³) / 12.
The parallel axis theorem states I = Ic + A × d², where Ic is the moment of inertia about the centroidal axis, A is the cross-sectional area, and d is the perpendicular distance between the centroidal axis and the new parallel axis.
I-beams place most of their material in the flanges, far from the neutral axis. Since the moment of inertia depends on the cube of the distance from the neutral axis, this geometry maximizes stiffness while minimizing weight, making I-beams highly efficient for bending resistance.
Moment of inertia has units of length to the fourth power (L⁴). Common units include mm⁴, cm⁴, m⁴ in metric, and in⁴ or ft⁴ in imperial. These units arise because the formula integrates area (L²) multiplied by distance squared (L²).

Frequently Asked Questions

This calculator computes Ix, the moment of inertia about the centroidal axis parallel to the base (horizontal axis). For symmetrical shapes, you can swap the base and height dimensions to obtain Iy.
The section modulus S = I / c relates moment of inertia to the maximum distance from the neutral axis (c). While I measures stiffness, S directly relates to bending stress: σ = M / S. Both are important in beam design.
No. All dimensions must be positive real numbers. For hollow sections, the inner dimensions must be strictly less than the outer dimensions. The calculator validates these conditions and will display appropriate error messages.
The I-beam formula assumes rectangular flanges and web with sharp corners, which is a standard engineering approximation. Real rolled steel sections have tapered flanges and fillets. For precise values, consult manufacturer tables or use finite element analysis.
Beam deflection is inversely proportional to EI (flexural rigidity). Doubling the moment of inertia halves the deflection under the same load and span conditions. This relationship is fundamental to the Euler-Bernoulli beam equation.
The radius of gyration r = √(I / A) is the distance from the axis at which the entire area could be concentrated to give the same moment of inertia. It is used extensively in column buckling analysis and is derived directly from I and the cross-sectional area A.

Moment of Inertia Glossary

Second Moment of Area

The geometric property I = ∫ y² dA that quantifies a cross-section's resistance to bending about a given axis. Synonymous with moment of inertia in structural engineering.

Centroidal Axis

An axis passing through the centroid (geometric center) of a cross-section. Moment of inertia is minimized about centroidal axes.

Neutral Axis

The axis within a beam cross-section where bending stress is zero. For symmetric sections under pure bending, this coincides with the centroidal axis.

Section Modulus

S = I / c, where c is the distance from the neutral axis to the extreme fiber. Used to calculate maximum bending stress: σ = M / S.

Radius of Gyration

r = √(I / A). A measure of how efficiently a cross-section distributes area relative to an axis. Essential for column buckling calculations.

Parallel Axis Theorem

I = Ic + Ad². Allows calculation of moment of inertia about any axis parallel to a centroidal axis by adding the Ad² transfer term.

Flexural Rigidity

EI, the product of elastic modulus and moment of inertia. Determines a beam's resistance to curvature under bending moment.

Polar Moment of Inertia

J = ∫ r² dA. Measures resistance to torsion (twisting) rather than bending. For circular sections, J = πd⁴/32.

Editorial Review & Methodology

This moment of inertia calculator was built and reviewed by the NumbrWiz Editorial Team with input from practicing structural engineers. All formulas are derived from classical mechanics and are cross-verified against standard engineering references including the AISC Steel Construction Manual and Eurocode design standards.

  • Formula verification: Each shape formula has been validated against closed-form solutions from authoritative structural mechanics textbooks.
  • Edge case testing: Tested with thin-walled sections, near-zero dimensions, and extreme aspect ratios to ensure numerical stability.
  • Engineering review: I-beam calculations verified against published section property tables for standard W-shapes and European IPE profiles.

Transparency note: All calculations run client-side in your browser. No data is ever collected, stored, or transmitted. Results are for educational and preliminary design purposes. For critical structural applications, verify using manufacturer-published section properties or consult a licensed structural engineer.

Page last reviewed: May 2026 · NumbrWiz Editorial Team