Rectangular sections
For a rectangle, the dimension perpendicular to the selected centroidal axis is raised to the third power in the formula. Rotating the section can significantly change its area moment.
Ix = bh³ / 12Engineering workspace · 001
Moment of Inertia Calculator.
Calculate the second moment of area for common cross-sections. Change dimensions, explore centroidal axes, and understand the mathematics behind each result.
Browser-based calculations Live results No account required
Open calculator Geometry input
Define your section
Solid rectangle
Horizontal dimension
Vertical dimension
Live analysis
Calculated properties
Solid rectangle
66,666,670
mm⁴
Iy16,666,670 mm⁴
Area20,000 mm²
Applied formula
Ix = bh³/12; Iy = hb³/12Length inputs are converted to millimetres for calculations. Review the assumptions before using results in engineering work.
The mechanics
What is the moment of inertia of an area?
The second moment of area describes how a cross-section's area is distributed relative to a chosen axis. It is used in beam bending and deflection calculations. Its value depends on both the cross-sectional shape and the axis being considered.
Circular sections
A solid circle has equal centroidal area moments about any two perpendicular diameters through its centre.
I = πd⁴ / 64Hollow sections
For concentric circular tubes, the inner circle's area moment is subtracted from the outer circle's area moment.
I = π(D⁴ − d⁴) / 64Method & units
How the calculator works
Select a supported cross-section, choose your input unit, and enter the required dimensions. The calculator converts lengths to millimetres before evaluating the formulas. Area moments are reported in mm⁴ and cross-sectional area in mm².
Choose a section
Select a rectangle, solid circle, hollow circle, or hollow rectangle from the cross-section menu.
Enter dimensions
Use positive dimensions and make sure inner dimensions are smaller than their corresponding outer dimensions.
Review the result
Inspect the x-axis and y-axis area moments, area, and the formula used for the selected section.
These are area moments of inertia, not mass moments of inertia. Composite sections, principal axes, product of inertia, and mass moment of inertia require separate calculations.
Calculator directory
Explore moment of inertia calculators
Select a calculator to explore its inputs, formulas, and results.
AREA
Rectangle Moment of Inertia Calculator
Calculate centroidal area moments for a rectangular cross-section using its width and height.
Open calculator → AREACircle Moment of Inertia Calculator
Calculate the centroidal area moment for a solid circular section from its diameter.
Open calculator → AREAHollow Rectangle Moment of Inertia Calculator
Explore area moments for hollow rectangular sections using outer and inner dimensions.
Open calculator → AREAHollow Circle Moment of Inertia Calculator
Calculate area moments for hollow circular and tube sections using inner and outer diameters.
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Calculate section properties for I-shaped cross-sections from their geometric dimensions.
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Explore area moments for T-shaped cross-sections and their centroidal axes.
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Calculate geometric section properties for channel-shaped cross-sections.
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Explore area moments for L-shaped angle sections using their dimensions.
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Calculate centroidal area moments for triangular cross-sections.
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Calculate area moments for semicircular cross-sections about defined axes.
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Calculate centroidal area moments for elliptical sections from their dimensions.
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Explore centroidal area moments for trapezoidal cross-sections.
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Calculate geometric section properties for Z-shaped cross-sections.
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Combine component shapes using centroid calculations and the parallel-axis theorem.
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Explore area properties for custom cross-sections defined by their geometry.
Open calculator → THEOREMSParallel Axis Theorem Calculator
Translate an area moment between parallel axes using the parallel-axis theorem.
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Calculate polar second moment of area for suitable cross-sections and axes.
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Calculate and understand the product of area inertia relative to defined axes.
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Calculate principal area moments from planar section properties.
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Explore rotational inertia for defined masses, geometries, and rotation axes.
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Calculate first area moments used in centroid and shear analysis.
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Calculate elastic section modulus from area moment and extreme-fibre distance.
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Calculate radius of gyration from area and second moment of area.
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Explore standard steel section properties when validated section data is available.
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