Moment of Inertia Calculator

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.

The area moment of inertia (also called the second moment of area or second moment of inertia) 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.

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

To find the second moment of the area when the origin of the coordinate system does not coincide with the centroid, use the parallel axes theorem. The moment of area about an axis parallel to the x-axis that lies in the distance a from it is given by the formula Ix + Aa², where:

• 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.

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

We can utilize the concept of the moment of inertia to analyze the cross-section of materials we use in construction to determine their strength.

If you want to learn how we can utilize the moment of inertia of a beam's cross-section, you can check out our beam deflection calculator or our wood beam span calculator. We also have our floor joist calculator if that interests you.