# Moment of Inertia Calculator

By Bogna Szyk
Last updated: Dec 02, 2020

If you have trouble with determining the second moment of area of any common shape (like a circle or a hexagon), this moment of inertia calculator is here to help you. You will also find the moment of inertia formulas here - read the description below to make sure you are using them correctly! We will explain how these formulas work so that you won't have to wonder how to calculate the moment of inertia of a rectangle ever again.

## Second moment of area

What exactly is the area moment of inertia (also called the second moment of area)? It is a geometrical property of any area. It describes how the area is distributed about an arbitrary axis. The units of the area moment of inertia are meters to the fourth power (m⁴).

We can distinguish between the moment of inertia about the horizontal x-axis (denoted `Ix`) and the moment of inertia about the vertical y-axis (denoted `Iy`). We usually assume that the "width" of any shape is the length of the side along the x-axis and the height - along the y-axis.

## Moment of inertia formulas

Generally, finding the second moment of area of an arbitrary shape requires integration. You can use the following equations for the most common shapes, though. Remember that these formulas are valid only if the origin of the coordinate system coincides with the centroid of the area. In other words, if both the x-axis and the y-axis cross the centroid of the analyzed shape, then these equations hold.

1. Triangle:

`Ix = width * height³ / 36`

`Iy = (height * width³ - height * a * width² + width * height * a²) / 36` where `a` is a top vertex displacement

2. Rectangle:

`Ix = width * height³ / 12`

`Iy = height * width³ / 12`

3. Circle:

`Ix = Iy = π/4 * radius⁴`

4. Semicircle

`Ix = [π/8 - 8/(9*π)] * radius⁴`

`Iy = = π/8 * radius⁴`

5. Ellipse:

`Ix = π/4 * radius_x * radius_y³`

`Iy = π/4 * radius_y * radius_x³`

6. Regular hexagon:

`Ix = Iy = 5*√(3)/16 * side_length⁴`

## How to calculate the moment of inertia about any axis

What if the origin of the coordinate system does not coincide with the centroid? Have no worries - it is still possible to find the second moment of the area! You need to use the parallel axes theorem. Let's say you want to find the moment of area about an axis parallel to x-axis that lies in distance `a` from it. You need to use the following formula:

`I = Ix + Aa²`

where:

• `I` is the moment of inertia about the axis parallel to the x-axis,
• `Ix` is the moment of inertia about the x-axis,
• `A` is the area, and
• `a` is the distance between two parallel axes.

To find the area of a circle (and its other properties as well), use the circumference calculator.

## Example: the moment of inertia of a rectangle

Let's assume a rectangle with width 12 cm and height 8 cm. Its centroid lies in the origin of the coordinate system. Then:

`Ix = 12 * 8³ / 12 = 512 cm⁴`

`Iy = 8 * 12³ / 12 = 1152 cm⁴`

Bogna Szyk
Shape
rectangle Width (b)
in
Height (a)
in
Ix
in⁴
Iy
in⁴
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