Moment of Inertia Calculator
This moment of inertia calculator applies the formula of moment of inertia to find the area moment of inertia of a beam section. Enter the shape and dimensions and the moment of inertia calculator returns the value, a key input for how much a beam bends. It uses the standard moment of inertia equation for rectangles and circles, so you get the answer without integrating by hand.
Results
The Moment of Inertia Calculator is part of our Force & Motion collection. Browse more force & motion to size, convert, and check your work alongside this tool.
What the formula of moment of inertia describes
The moment of inertia measures how a cross-section resists bending, and the formula of moment of inertia depends entirely on the shape and how its material is spread from the neutral axis. This moment of inertia calculator works it out for the common beam sections. Material far from the axis contributes far more, which is why an I-beam is so stiff for its weight, and why the area moment of inertia formula squares the distance.
The formula of moment of inertia
For a rectangle bending about its horizontal axis, the formula of moment of inertia is:
where b is the width and h is the height. For a solid circle the moment of inertia equation is I = π × d⁴ ÷ 64. In both, the dimension in the bending direction is raised to a high power, which is why depth matters so much in a beam. The moment of inertia calculator evaluates the right formula for the shape you pick, so the area moment of inertia formula never has to be worked by hand.
Formula of moment of inertia example
A rectangular section 50 mm wide and 100 mm deep. Using the formula of moment of inertia:
That is about 417 cm⁴. Turn the section on its side, 100 wide by 50 deep, and I drops to about 1,041,667 mm⁴, a quarter as stiff. Enter your section and the moment of inertia calculator shows the difference.
How to use the moment of inertia calculator
Choose the section shape, then enter the dimensions it asks for: width and height for a rectangle, or diameter for a circle. The moment of inertia calculator applies the formula of moment of inertia for that shape and returns the value in mm⁴ and cm⁴. As an area moment of inertia calculator it gives the figure beam-bending and deflection formulas need next.
Reading the moment of inertia calculator result
A larger moment of inertia means a stiffer section that bends less under the same load. The value feeds directly into beam deflection and bending stress, where more inertia means less deflection and lower stress. Because depth is cubed in the area moment of inertia formula, orienting a section so its tall dimension resists the load is the cheapest way to add stiffness. The moment of inertia calculator gives the number; the beam formulas turn it into a deflection.
When to use the moment of inertia calculator
- Sizing a beam with the formula of moment of inertia
- Feeding deflection formulas with the area moment of inertia calculator
- Comparing section orientations with the moment of inertia formula rectangle
- Checking a design with the second moment of area calculator
Common moment of inertia equation mistakes
- Wrong axis. The formula of moment of inertia depends on the bending direction; a tall section on its side is far weaker.
- Area versus polar. Use area moment of inertia for bending, the polar moment of inertia formula for torsion.
- Unit slips. The result is in mm⁴; keep every dimension in millimeters.
Tips for accurate results
- Enter the depth in the bending direction so the formula of moment of inertia uses the right axis.
- Add depth, not width, to stiffen a beam efficiently.
- Use the area value for bending and the polar value for torsion.
Why depth dominates the formula of moment of inertia
The most important lesson in the formula of moment of inertia is the power on the depth. In a rectangle, height is cubed, so doubling the depth of a beam multiplies its stiffness by eight, while doubling the width only doubles it. This is why joists stand tall rather than lie flat, and why adding depth is the most material-efficient way to stiffen a beam. The area moment of inertia formula rewards moving material away from the neutral axis, which is the whole idea behind I-beams: the flanges sit far from the center where they do the most good, and the web just holds them apart.
Area versus polar moment of inertia formula
There are two moments of inertia that share a name, and mixing them up is a common error. The area moment of inertia, what this moment of inertia calculator finds, governs bending and appears in deflection and stress formulas. The polar moment of inertia formula governs torsion, how a shaft resists twisting, and uses the distance from the center in all directions rather than from one axis. For a circle the polar value is exactly twice the area value. When you size a beam for bending you want the area moment of inertia; when you size a shaft for torque you want the polar one, so pick the quantity that matches the load.
Formula of moment of inertia chart
The formula of moment of inertia for common shapes about the horizontal axis:
| Shape | Formula | Example value |
|---|---|---|
| Rectangle | b·h³ ÷ 12 | 50×100: 4.17×10⁶ mm⁴ |
| Circle | π·d⁴ ÷ 64 | d50: 3.07×10⁵ mm⁴ |
| Hollow circle | π(D⁴−d⁴) ÷ 64 | depends on wall |
| Square | a⁴ ÷ 12 | a50: 5.21×10⁵ mm⁴ |
Moment of Inertia Calculator FAQ
What is the formula of moment of inertia?
For a rectangle it is I = b × h³ / 12; for a circle, I = π × d⁴ / 64. The moment of inertia calculator applies the right area moment of inertia formula for the shape you choose.
What are the units?
The area moment of inertia is in length to the fourth power, mm⁴ or cm⁴. The moment of inertia calculator reports both.
Why does orientation matter?
Because height is cubed in the moment of inertia formula rectangle, a section is far stiffer standing tall than lying flat. Turning it on its side can cut the inertia to a quarter.
Is this the same as rotational inertia?
This is the area moment of inertia for bending. The rotational inertia formula for a spinning mass is a related but different quantity, mass times radius squared.