Moment of Inertia of a Rectangle

Moment of Inertia of a Rectangle — Formula and Worked Example

The rectangle is the reference case every other section is measured against. Its second moment of area about the centroidal axis is bh³/12 — note the cube on the depth, which is why orienting a joist on edge rather than flat is so effective.

Formulas

The depth term is cubed, so doubling h multiplies Iₓ by eight while doubling b only doubles it. Note also the base value bh³/3: that is the parallel-axis result Iᶜ + A(h/2)², and it is four times the centroidal value — a common source of error if you use the wrong one.

Worked Example — A 40 × 60 mm rectangle

Values computed by the calculator above. Change any dimension and every property updates live.

What is the moment of inertia of a rectangle?

About the centroidal axis parallel to the base it is Ix = bh³/12, where b is the width and h the depth. About the base itself it is bh³/3, which follows from the parallel axis theorem.

Why is a joist stronger on edge than laid flat?

Because Ix depends on the cube of the depth. A 50 × 150 mm joist on edge has Ix = 50 × 150³/12 = 14.1 × 10⁶ mm⁴, but laid flat it is 150 × 50³/12 = 1.56 × 10⁶ mm⁴ — nine times less stiff for exactly the same material.

What is the difference between bh³/12 and bh³/3?

bh³/12 is about the centroidal axis through mid-depth; bh³/3 is about the base. They differ by the transfer term A(h/2)² = bh³/4 from the parallel axis theorem. Use the centroidal value for bending stress unless you specifically need the base.

Where bh³/12 Comes From

The second moment of area is defined as Iₓ = ∫y² dA. For a rectangle of width b, a horizontal strip at height y has area dA = b dy, so integrating from −h/2 to +h/2 gives:

Iₓ = ∫_−h/2^h/2 y² b dy = b[y³/3]_−h/2^h/2 = bh³/12

The cube on h falls straight out of integrating y². Integrating instead from 0 to h — that is, about the base rather than the centroid — gives bh³/3, four times larger.

Orientation Matters: On Edge vs Laid Flat

The same piece of material, rotated 90°, gives a completely different stiffness. Iₓ values in 10⁶ mm⁴:

The ratio is simply (h/b)², so a section twice as deep as it is wide is four times stiffer on edge, and a 50 × 200 is sixteen times stiffer. This is the single most important practical consequence of the h³ term.

Moment of Inertia of 2x4, 2x6, 2x8, 2x10 and 2x12 Lumber

US dimension lumber is named by its rough size but planed smaller, so the calculation uses the actual dressed size (a 2x6 is 1.5 × 5.5 in). About the strong axis, with the board on edge:

Worked for the 2x6: Iₓ = 1.5 × 5.5³ / 12 = 1.5 × 166.375 / 12 = 20.80 in⁴. Switch the calculator to Imperial and enter b = 1.5, h = 5.5 to check it.

What is the difference between Ix and Iy of a rectangle?

Iₓ = bh³/12 is about the horizontal centroidal axis and Iₖ = hb³/12 about the vertical one. The cubed dimension is always the one measured perpendicular to the axis, so for a 2x6 on edge Iₓ is 20.80 in⁴ while Iₖ is only 1.55 in⁴.

What happens to the moment of inertia if you double the height?

It goes up eight times, because h is cubed. A 50 × 100 mm rectangle has Iₓ = 4.167 × 10⁶ mm⁴; at 50 × 200 mm it is 33.33 × 10⁶ mm⁴. Doubling the width instead only doubles it.

Is the moment of inertia of a rectangle the same as its mass moment of inertia?

No, and they are not interchangeable. bh³/12 is the area moment (units mm⁴) and governs bending stiffness. The mass moment of inertia of a rectangular plate about its centroidal axis is m(h²+t²)/12 (units kg·m²) and governs rotational acceleration. Same name, different quantity — this calculator computes the area moment.

Other Cross-Sections

The same calculator handles all eight standard sections:

• I-Beam
• Channel
• Angle
• T-Section
• Rectangle (this page)
• Circle
• Hollow Circle
• Hollow Rect

For the full section-property reference and the parallel-axis theorem, use the moment of inertia calculator for any cross-section.

Updated 3 October 2026