Area Moment of Inertia Calculator (Rectangle, Circle, Tube)
By Santaji GadepatilinUpdated November 3, 2026
The second moment of area I describes how a cross-section's material is spread around its bending axis, and it decides how stiff a beam is. This area moment of inertia calculator gives I and the elastic section modulus Z for a rectangle and for a solid or hollow circular section, in the cm⁴ and cm³ units used in steel tables.
How to use the area moment of inertia calculator
For a rectangle, enter the width b and the depth h measured in the direction of bending. For a round section, enter the outer diameter and the inner diameter (0 for a solid bar). The calculator returns the second moment of area I in cm⁴ and the elastic section modulus Z in cm³ for both shapes — ready for the beam deflection and bending stress calculators.
Second moment of area formulas
Rectangle: I = b h³ ÷ 12 · Z = b h² ÷ 6
Circle / tube: I = π (D⁴ − d⁴) ÷ 64 · Z = π (D⁴ − d⁴) ÷ (32 D)
1 cm⁴ = 10⁻⁸ m⁴ and 1 cm³ = 10⁻⁶ m³.
Example
A 100 × 200 mm timber beam bending about its strong axis, and a 60 mm solid steel bar:
- Rectangle: I = 100 × 200³ ÷ 12 = 66,666,667 mm⁴ = 6,666.7 cm⁴; Z = 100 × 200² ÷ 6 = 666.7 cm³
- Solid bar: I = π × 60⁴ ÷ 64 = 63.62 cm⁴; Z = π × 60³ ÷ 32 = 21.21 cm³
Rectangular sections
| b × h | I (strong axis) | Z (strong axis) | I (weak axis) |
|---|---|---|---|
| 50 × 100 mm | 416.7 cm⁴ | 83.3 cm³ | 104.2 cm⁴ |
| 50 × 150 mm | 1,406.2 cm⁴ | 187.5 cm³ | 156.2 cm⁴ |
| 75 × 150 mm | 2,109.4 cm⁴ | 281.2 cm³ | 527.3 cm⁴ |
| 100 × 200 mm | 6,666.7 cm⁴ | 666.7 cm³ | 1,666.7 cm⁴ |
| 100 × 300 mm | 22,500.0 cm⁴ | 1,500.0 cm³ | 2,500.0 cm⁴ |
| 150 × 300 mm | 33,750.0 cm⁴ | 2,250.0 cm³ | 8,437.5 cm⁴ |
Steel pipe (Schedule 40) as a beam
| Pipe | I | Z |
|---|---|---|
| 25 NB (33.4 × 3.4) | 3.6 cm⁴ | 2.18 cm³ |
| 50 NB (60.3 × 3.9) | 27.7 cm⁴ | 9.18 cm³ |
| 80 NB (88.9 × 5.5) | 125.7 cm⁴ | 28.27 cm³ |
| 100 NB (114.3 × 6.0) | 301.1 cm⁴ | 52.68 cm³ |
| 150 NB (168.3 × 7.1) | 1,171.6 cm⁴ | 139.23 cm³ |
| 200 NB (219.1 × 8.2) | 3,018.7 cm⁴ | 275.55 cm³ |
Notes
- Turning a rectangle on its side swaps b and h: a 100 × 200 mm plank is four times stiffer on edge than flat.
- For built-up shapes, add the I of each part about the common axis using the parallel-axis theorem, I = Ic + A d².
Frequently asked questions
What is the moment of inertia of a rectangle?
I = b h³ ÷ 12 about the centroidal axis parallel to b, where h is the depth in the direction of bending. The section modulus is Z = b h² ÷ 6.
What is the moment of inertia of a circle and a tube?
Solid circle I = π d⁴ ÷ 64; hollow tube I = π (D⁴ − d⁴) ÷ 64. The section modulus is I divided by the outer radius, Z = π (D⁴ − d⁴) ÷ (32 D).
Is area moment of inertia the same as mass moment of inertia?
No. The area moment (m⁴) describes a cross-section's resistance to bending; the mass moment (kg·m²) describes a body's resistance to rotation. They share a name and the r² weighting but are used for different problems.
Why are I-beams so efficient?
I grows with the square of the distance from the neutral axis, so putting most material in flanges far from the centre gives a high I and Z for little weight.