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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.

What is the area moment of inertia?

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.

🙋 Want to learn about the mass moment of inertia instead, which we can express in kilogram-square meters (kg·m²) or pound-square foot (lb·ft²)? Then our mass moment of inertia calculator is what you need to check.

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 do I calculate the moment of inertia about any axis?

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.

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

Different applications of the moment of inertia

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.

FAQ

How do I calculate the area moment of inertia of a composite shape?

Area moment of inertia is additive, which means that the moment of area for a complicated shape is the sum of the area moment of inertia of all of its constituents. If there is a "hole", you simply subtract its area moment of inertia (instead of adding it).

What is the area moment of inertia of a circle with radius 4?

The answer is 201.06. To arrive at this result, you need to remember the area moment of inertia formula for a circle: r⁴ × π / 4, where r is the radius of your circle.

How do I calculate the area moment of inertia of an annulus?

To determine the area moment of inertia of an annulus:

  1. Determine the outside and inside radii of your annulus. Let's denote them by R and r, respectively.

  2. Compute the area moment of inertia of a circle with radius R. Recall the formula is radius⁴ × π / 4.

  3. Compute the area moment of inertia of a circle with radius r.

  4. Subtract the result in Step 3 from the result in Step 2.

  5. That's it! We've used the additivity of the area moment of inertia.

Bogna Szyk
Shape
Rectangle
Moment of inertia of a rectangle
Width (b)
in
Height (a)
in
Ix
in⁴
Iy
in⁴
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