# Moment of Inertia Calculator

URL: https://speedycalculator.com/calculators/math/moment-of-inertia/
Last updated: 2026-10-30

> The moment of inertia calculator finds the second moment of area Iₓ and I_y about the centroidal axes, and the polar moment J, for a rectangle, triangle, circle, semicircle, ellipse or regular hexagon.

## Inputs

- Shape
- Base (width) b (cm, m, mm, in, ft, yd, km, mi)
- Height h (cm, m, mm, in, ft, yd, km, mi)
- Radius r (cm, m, mm, in, ft, yd, km, mi)
- Semi-axis a (horizontal) (cm, m, mm, in, ft, yd, km, mi)
- Semi-axis b (vertical) (cm, m, mm, in, ft, yd, km, mi)
- Side s (cm, m, mm, in, ft, yd, km, mi)

## Results

- Iₓ (horizontal centroidal axis)
- I_y (vertical centroidal axis)
- Polar moment J = Iₓ + I_y

## 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
```

## Explanation

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

FieldWhat to enterShapeChoose one: Rectangle, Triangle, Circle, Semicircle …Base (width) bUnits: cm, m, mm, in, ft, yd …Height hUnits: cm, m, mm, in, ft, yd …Radius rUnits: 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 sUnits: 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)53,333.33337.55,000106,666.666712.58,333.33331510,0002013,333.3333

## 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.

## FAQ

**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.


## Sources

- [Wolfram MathWorld – Area moment of inertia](https://mathworld.wolfram.com/AreaMomentofInertia.html)
- [OpenStax – Calculus Volume 2](https://openstax.org/details/books/calculus-volume-2)