Engineering Reference

Moment of Inertia Calculator

Compute the second moment of area (moment of inertia) for a rectangular or circular cross-section, with the radius of gyration.

Page last changed 2026-10-19 · based on n/a — standard engineering relationships, no single governing revision

Quick Answer

For a rectangle, I = b·h³ ÷ 12 about the axis through its centroid — note the height is cubed, so doubling the depth multiplies stiffness by eight. For a solid circle, I = π·d⁴ ÷ 64.

Section Moment of Inertia

The Formulas Used

Rectangle about the centroid: Ix = b·h³ ÷ 12   Iy = h·b³ ÷ 12
Solid circle: I = π·d⁴ ÷ 64   A = π·d² ÷ 4   r = d ÷ 4
Radius of gyration: r = √(I ÷ A)

Where the Formula Comes From

The second moment of area is the integral of y² over the section, where y is the distance from the axis. For a rectangle of width b and height h, integrating y² from −h/2 to +h/2 gives b·h³/12 — the cube on the height is the whole story of why depth beats width in bending.

A 2 × 4 laid flat has Ix = 10.67 in⁴. Stand it on edge and Ix becomes 21.3 in⁴ — twice the stiffness from the same piece of timber, purely because the depth went from 2 to 4 in. That is why floor joists are set on edge and why depth, not area, is what you increase when a floor is bouncy.

What the Calculator Does Not Cover

Only a solid rectangle and a solid circle are handled here. Hollow sections, I-beams, channels and angles have published section property tables, and the built-up sections in structural design — T-sections, composite beams — need the parallel axis theorem, which this calculator does not apply.

For an I-section, use the section properties chart. Note also that this is the second moment of area (a geometric property, in in⁴) — not the mass moment of inertia (in lb·in·s²), which is a different quantity used in dynamics.

Frequently Asked Questions

What is the moment of inertia of a rectangle?
I = b·h³/12 about the centroidal axis parallel to the width. For a 2 in × 4 in rectangle that is 2 × 64 / 12 = 10.67 in⁴ about the axis through the middle, with the 4 in dimension acting as the depth.
Why is it called the second moment of area and not the moment of inertia?
Because it is the second moment of the area about an axis — a purely geometric quantity with units of length to the fourth power. Mass moment of inertia, used in dynamics, has units of mass × length² and describes resistance to angular acceleration. The names overlap and the quantities do not.
Does doubling the depth double the stiffness?
No — it multiplies it by eight. Stiffness in bending is proportional to I, and I is proportional to the cube of the depth. Going from a 2 in to a 4 in deep beam is 2³ = 8 times the stiffness, assuming the same width and material.
How do I find the moment of inertia for an I-beam?
Use a published section property table rather than computing it — I-beam tables list Ix, Iy, Sx and rx directly. An I-section can be computed as the difference of two rectangles, but the fillets and the standard's rounding make the published values the ones to design from.
What is the radius of gyration?
r = √(I/A), the distance at which the entire area could be concentrated to give the same I. It is the quantity that appears in column buckling formulas, and a section with a larger radius of gyration resists buckling better for the same area.

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