A rectangle is the most common section in structural design: concrete beams, timber joists and plates. Its moment of inertia about the centroidal axis grows with the cube of the depth, which is why beams are deeper than they are wide.
Formula
Worked example (millimetres)
Given
- Width b = 300 mm
- Depth h = 500 mm
Steps
- Strong axis: Iₓ = b × h³ / 12 = 300 × 500³ / 12 = 3,125,000,000 mm⁴ (312,500 cm⁴).
- Weak axis: I_y = h × b³ / 12 = 500 × 300³ / 12 = 1,125,000,000 mm⁴ (112,500 cm⁴).
- Section modulus: Zₓ = Iₓ / (h/2) = b × h² / 6 = 12,500,000 mm³.
- Radius of gyration: rₓ = h / √12 = 144.3 mm.
| Area, A | 150,000 mm² |
|---|---|
| Moment of inertia, Iₓ | 3,125,000,000 mm⁴ |
| Section modulus, Zₓ | 12,500,000 mm³ |
| Radius of gyration, rₓ | 144.3 mm |
Worked example (inches)
Given
- Width b = 6 in
- Depth h = 12 in
Steps
- Strong axis: Iₓ = b × h³ / 12 = 6 × 12³ / 12 = 864 in⁴.
- Weak axis: I_y = h × b³ / 12 = 12 × 6³ / 12 = 216 in⁴.
- Section modulus: Zₓ = Iₓ / (h/2) = b × h² / 6 = 144 in³.
- Radius of gyration: rₓ = h / √12 = 3.46 in.
| Area, A | 72 in² |
|---|---|
| Moment of inertia, Iₓ | 864 in⁴ |
| Section modulus, Zₓ | 144 in³ |
| Radius of gyration, rₓ | 3.46 in |
Try your own dimensions
The free calculator gives area, centroid, Iₓ, I_y, section modulus and radius of gyration in mm or inches.
Open the Moment of Inertia Calculator →Frequently asked questions
- What is the moment of inertia of a rectangle?
- About its centroidal axis parallel to the width, Iₓ = b × h³ / 12, where b is the width and h the depth. About the other axis, I_y = h × b³ / 12.
- Why is the depth cubed in the rectangle formula?
- Each strip of area contributes area × distance², and the strips get further from the axis as the section gets deeper. Integrating over the depth gives h³. Doubling the depth makes the beam eight times stiffer; doubling the width only doubles it.
- What is the moment of inertia of a rectangle about its base?
- About an axis along one edge, I = b × h³ / 3. This follows from the parallel-axis theorem: b × h³ / 12 + (b × h) × (h/2)².