Moment of Inertia Calculator

The area moment of inertia — also called the second moment of area — is a purely geometric property that describes how a cross-section's area is distributed relative to a given axis. It has units of length to the fourth power (e.g. mm⁴ or m⁴) and is widely used in structural and mechanical engineering to predict how a beam will resist bending and deflection.

Ix is the second moment of area about the horizontal centroidal x-axis, and Iy is the second moment of area about the vertical centroidal y-axis. Ix governs bending resistance in the vertical plane (the most common loading direction for beams), while Iy governs resistance to lateral bending.

For a solid rectangle of width b and height h, Ix = (b × h³) / 12 and Iy = (h × b³) / 12, both measured about the centroidal axes. The centroid lies at h/2 from the bottom and b/2 from the side. These are among the most commonly used formulas in structural engineering.

The parallel axis theorem states that I = I_c + A × d², where I_c 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 axis. This allows you to combine individual shapes into composite sections.

Divide the composite shape into simple sub-shapes (rectangles, circles, etc.) whose individual moments of inertia are known. Find the centroid of the whole section, then apply the parallel axis theorem to shift each sub-shape's Ix or Iy to the overall centroidal axis, and sum them all up. Subtract the inertia of any void or hole regions.

The section modulus S = I / y_max, where y_max is the distance from the centroid to the extreme fibre. It directly relates bending moment to bending stress via σ = M / S. A larger section modulus means the cross-section can carry higher bending moments before reaching the material's allowable stress.

For a solid circle of radius r, Ix = Iy = π × r⁴ / 4. For a hollow circular annulus with outer radius r and inner radius r_i, Ix = Iy = π × (r⁴ − r_i⁴) / 4. Circular sections are commonly used for columns, shafts, and pipes where bending may occur in any direction.

Engineers use the second moment of area to design beams, columns, floor joists, and structural members so they resist bending without excessive stress or deflection. It is also essential when selecting standard steel sections (I-beams, channels, hollow sections) from design tables, and when verifying compliance with structural codes such as Eurocode 3 or AISC.