The second moment of area measures how a cross-section resists bending: deflection and bending stress are inversely proportional to it. That is why I-beams put material far from the centre – a tall section is much stiffer than a flat one of the same area.
How to use the Moment of Inertia Calculator
- Shape
- Base (width) b
- Height h
- Radius r
- Semi-axis a (horizontal)
- Semi-axis b (vertical)
- Side s
Results update as you type, and everything is calculated in your browser.
How it works
Choose the shape and enter its dimensions. Iₓ is about the horizontal axis through the centroid, I_y about the vertical axis. The triangle is taken as isosceles for I_y.
Inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose one: Rectangle, Triangle, Circle, Semicircle … |
| Base (width) b | Units: cm, m, mm, in, ft, yd … |
| Height h | Units: cm, m, mm, in, ft, yd … |
| Radius r | Units: cm, m, mm, in, ft, yd … |
| Semi-axis a (horizontal) | Units: cm, m, mm, in, ft, yd … |
| Semi-axis b (vertical) | Units: cm, m, mm, in, ft, yd … |
| Side s | Units: cm, m, mm, in, ft, yd … |
Example values
How the main result changes when Base changes and everything else stays at the example values above:
| Base (width) b (cm) | Iₓ (horizontal centroidal axis) (cm⁴) |
|---|---|
| 5 | 3,333.3333 |
| 7.5 | 5,000 |
| 10 | 6,666.6667 |
| 12.5 | 8,333.3333 |
| 15 | 10,000 |
| 20 | 13,333.3333 |
Formula
Rectangle Iₓ = bh³/12 · Triangle Iₓ = bh³/36 · Circle Iₓ = πr⁴/4
Semicircle Iₓ = (π/8 − 8/(9π))r⁴ · Ellipse Iₓ = πab³/4 · Hexagon Iₓ = 5√3/16·s⁴ · J = Iₓ + I_y
Worked example
Inputs — Shape: Rectangle; Base (width) b: 10 cm; Height h: 20 cm.
Results — Iₓ (horizontal centroidal axis) = 6,666.666667 cm⁴; I_y (vertical centroidal axis) = 1,666.666667 cm⁴; Polar moment J = Iₓ + I_y = 8,333.333333 cm⁴.
Key terms
- Second moment of area
- ∫y² dA, the bending stiffness of a section.
Frequently asked questions
Is this the same as mass moment of inertia?
No – this is the area moment (units length⁴) used in beam bending.
Why are I-beams so efficient?
Most of their area is far from the neutral axis, where it contributes most to I.
How do I shift the axis?
Parallel axis theorem: I = I_c + A·d².
What is the polar moment used for?
Torsion of shafts: twist angle ∝ 1/J.