Area moment vs mass moment: two different things

The phrase “moment of inertia” is used for two unrelated quantities, which is why this tool has two modes.

The area moment of inertia (also called the second moment of area, symbol I, units of length to the fourth power) describes how a cross-section’s material is spread about an axis. It governs how stiff a beam is in bending and how much it deflects. A taller section has a much larger area moment because the height is cubed.

The mass moment of inertia (symbol I, units of mass times length squared) describes how a body’s mass is spread about a rotation axis. It is the rotational equivalent of mass: it sets how much torque you need to angularly accelerate a shaft, wheel, or flywheel. Use the mass mode for anything that spins.

What this calculator reports

In area mode you get the moment about both principal axes (Ix and Iy), the polar moment J = Ix + Iy for torsion, the section modulus S = I ÷ c for bending stress, the radius of gyration r = sqrt(I ÷ A) for buckling, the centroid location, and the cross-sectional area. In mass mode you get the inertia in both kg·m² and lb·ft², the radius of gyration, the mass used, and, if you enter an RPM, the rotational kinetic energy.

The parallel axis theorem, explained

Formulas give the moment of inertia about an axis through the centroid. To find it about any parallel axis a distance d away, add the area (or mass) times d squared:

This is how the calculator builds I-beams, tees, channels, angles, and any custom composite: it finds each piece’s own moment, shifts it to the shared centroid with the parallel axis theorem, and adds them up. The optional offset field lets you apply it directly to any shape.

Common section properties

Area moment of inertia about the centroidal axis for everyday shapes:

Mass moment of inertia for common rotating bodies (m is mass, r is radius, L is length):

Worked steel examples

A 4 in square steel tube with a 0.25 in wall has Ix = (4·4³ − 3.5·3.5³) / 12 = 8.81 in⁴, far stiffer than a solid 2 in bar (1.33 in⁴) for a fraction of the weight, because the material sits far from the neutral axis. A steel flywheel 300 mm across with a 200 mm bore and a mass of 20 kg has a mass moment of about 0.41 kg·m² on its axis, the figure that sets how much torque spins it up.

Moment of inertia units

Area moment has units of length to the fourth power: in⁴, ft⁴, mm⁴, cm⁴, or m⁴. Be careful converting: 1 in⁴ equals 416231 mm⁴, because the inch-to-millimetre factor of 25.4 is raised to the fourth power. Section modulus is length cubed and radius of gyration is a length. Mass moment uses kg·m² in SI or lb·ft² in US units, where 1 kg·m² equals 23.73 lb·ft².

Moment of inertia for beams

In a beam, bending stiffness is E·I and deflection scales with 1/I, so doubling the area moment halves the deflection under the same load. Because the height term is cubed, making a beam taller helps far more than making it wider. Take the Ix from this calculator straight into a deflection or stress check; the section modulus Sx = Ix ÷ c is what turns a bending moment into a stress.

Moment of inertia for shafts

For a rotating or twisted shaft the property that matters is the polar moment J. For a solid round shaft J = π·d⁴/32 and for a hollow shaft J = π·(D⁴ − d⁴)/32. Torsional shear stress is τ = T·r ÷ J, so a larger J means less twist and lower stress for the same torque. Note that J = Ix + Iy is the true polar moment only for solid and hollow round sections; open shapes warp and need a separate torsion constant.

Frequently asked questions