Moment of Inertia of a Rectangle
Iₓ = bh³/12 • Section Modulus & Radius of Gyration
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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
| Quantity | Expression |
|---|---|
| Area | A = bh |
| Second moment about x | Iₓ = bh³ / 12 |
| Second moment about y | Iₖ = hb³ / 12 |
| Section modulus | Sₓ = bh² / 6 |
| Radius of gyration | rₓ = h / √12 = 0.2887h |
| About the base (not centroid) | Iₛₐₛₑ = bh³ / 3 |
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
| Quantity | Value |
|---|---|
| b (width) | 40 mm |
| h (depth) | 60 mm |
| A | 2,400 mm² |
| Iₓ = bh³/12 | 720 × 10³ mm⁴ |
| Iₖ = hb³/12 | 320 × 10³ mm⁴ |
| Sₓ = bh²/6 | 24.0 × 10³ mm³ |
| rₓ = h/√12 | 17.3 mm |
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/2h/2 y² b dy = b[y³/3]−h/2h/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:
| Section (mm) | Iₓ on edge | Iₓ laid flat | Stiffness ratio |
|---|---|---|---|
| 50 × 100 | 4.17 | 1.04 | 4× |
| 50 × 150 | 14.06 | 1.56 | 9× |
| 50 × 200 | 33.33 | 2.08 | 16× |
| 75 × 150 | 21.09 | 5.27 | 4× |
| 100 × 200 | 66.67 | 16.67 | 4× |
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.
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, see the Moment of Inertia Calculator hub.