Is this the same as polar moment of inertia?

No. This page calculates area moments Ix and Iy for bending about centroidal axes. A polar moment is normally used for torsion in circular shafts and is often written as J. For a solid circle, J equals Ix + Iy, but for general open sections that shortcut is not a complete torsion design method.

Which moment of inertia should I use for beam deflection?

Use the moment of inertia about the axis that matches the bending direction. For a vertical load on a normally oriented rectangular joist, the strong-axis value Ix is usually used because the depth is vertical. If the member is rotated, the relevant axis changes.

Why does height affect Ix so strongly?

The rectangle formula Ix = b h³ / 12 contains the depth cubed. Doubling the depth increases Ix by eight times if width is unchanged. This is the reason deeper beams are much stiffer in bending.

Can I use this for an actual steel member design?

You can use the results for preliminary section-property checks, but final steel design should use published section properties, code checks for local buckling, lateral torsional buckling, shear, bearing, connections, and the applicable safety format.

Why is the hollow rectangle formula subtractive?

A centered hollow section can be treated as a large solid rectangle minus the missing rectangular opening. Because both areas share the same centroidal axes, the subtraction is direct. If the hole is off-center, the parallel axis theorem must be applied to both areas.

Engineering

Moment of Inertia Calculator

Calculate the second moment of area for common engineering sections and see the area, centroid, section modulus, and radius of gyration needed for beam stress and deflection checks.

Choose a section shape, enter dimensions in one consistent unit, and calculate centroidal Ix, Iy, section modulus, and radius of gyration. This is an area moment of inertia calculator, not a rotating-mass inertia calculator.

Shape

Unit label

Section property result

—

Area and centroid—

Ix / Iy—

Section modulus—

Radius of gyration—

—

What does this moment of inertia calculator measure?

In beam design, “moment of inertia” normally means the second moment of area, not the mass moment of inertia of a rotating body. It measures how far the cross-sectional area is distributed from an axis, so a deeper section can be much stiffer than a shallow section with the same area.

When the bending axis is not through the centroid, the value must be shifted using the parallel axis theorem. That is why the calculator separates centroidal section properties from the design use of those properties.

Worked example: rectangular beam section

For b = 100 mm and h = 200 mm: A = 100 × 200 = 20,000 mm² Ix = b h³ / 12 = 100 × 200³ / 12 = 66,666,666.7 mm⁴ Sx = Ix / (h/2) = 666,666.7 mm³

The section modulus is often the more directly useful value for bending stress because elastic bending stress can be estimated as σ = M / S. Deflection checks still need E × I, so the moment of inertia remains critical.

Common mistakes when using area moment of inertia

Do not mix millimetres and inches in the same calculation. Inertia uses the fourth power of length, so a unit mistake becomes extremely large. Also check the bending direction: Ix is not interchangeable with Iy unless the section is symmetric in both directions, such as a solid circle or round tube.

Common questions

No. This page calculates area moments Ix and Iy for bending about centroidal axes. A polar moment is normally used for torsion in circular shafts and is often written as J. For a solid circle, J equals Ix + Iy, but for general open sections that shortcut is not a complete torsion design method.
Use the moment of inertia about the axis that matches the bending direction. For a vertical load on a normally oriented rectangular joist, the strong-axis value Ix is usually used because the depth is vertical. If the member is rotated, the relevant axis changes.
The rectangle formula Ix = b h³ / 12 contains the depth cubed. Doubling the depth increases Ix by eight times if width is unchanged. This is the reason deeper beams are much stiffer in bending.
You can use the results for preliminary section-property checks, but final steel design should use published section properties, code checks for local buckling, lateral torsional buckling, shear, bearing, connections, and the applicable safety format.
A centered hollow section can be treated as a large solid rectangle minus the missing rectangular opening. Because both areas share the same centroidal axes, the subtraction is direct. If the hole is off-center, the parallel axis theorem must be applied to both areas.