Moment of Inertia Calculator
A moment of inertia calculator uses I = width × height³ ÷ 12 for a rectangle: a 1.5 × 9.25 in board has I of 98.93 in⁴ and a section modulus of 21.39 in³.
What the moment of inertia calculator finds
A moment of inertia calculator measures how well a cross-section resists bending. Enter the width and height of a rectangular section in inches. The tool returns the moment of inertia I in inches to the fourth power and the section modulus S in cubic inches.
Picture a plank. Lay it flat across two supports and it sags under your weight. Stand the same plank on its edge and it barely flexes. Same wood, same amount of it. What changed is I, and the height is cubed.
The formulas in plain words
- I = width × height × height × height ÷ 12.
- S = width × height × height ÷ 6.
Height is measured in the direction the load pushes. These apply to a solid rectangle bending about the axis through its center.
Because height is cubed in I and squared in S, a small gain in height pays off a lot.
Worked example: a 2x10 joist on edge
Actual size 1.5 in wide by 9.25 in tall.
- I: 1.5 × 9.25³ ÷ 12 = 1.5 × 791.45 ÷ 12 = 98.93 in⁴
- S: 1.5 × 9.25² ÷ 6 = 1.5 × 85.5625 ÷ 6 = 21.39 in³
Worked example: a 4x6 post turned both ways
Actual size 3.5 in by 5.5 in.
- With 5.5 as the height: I = 3.5 × 5.5³ ÷ 12 = 48.53 in⁴ and S = 17.65 in³
- With 3.5 as the height: I = 5.5 × 3.5³ ÷ 12 = 19.65 in⁴ and S = 11.23 in³
Same timber, but the strong way has about 2.5 times the stiffness.
Properties of common lumber sizes on edge
| Nominal size | Actual (width × height) | I (in⁴) | S (in³) |
|---|---|---|---|
| 2x4 | 1.5 × 3.5 | 5.36 | 3.06 |
| 2x6 | 1.5 × 5.5 | 20.80 | 7.56 |
| 2x8 | 1.5 × 7.25 | 47.63 | 13.14 |
| 2x10 | 1.5 × 9.25 | 98.93 | 21.39 |
| 2x12 | 1.5 × 11.25 | 177.98 | 31.64 |
Compare the 2x6 to the 2x12. The 2x12 is about twice as tall but has about 8.6 times the I. That is the cube doing its work.
Where the numbers go next
Once you have I and S, you feed them into other checks. S goes into the bending stress calculator to turn a moment into stress. I goes into the beam deflection calculator or beam span calculator to see how far a beam sags.
Those tools need a material stiffness, a load, and a span as well. The section is only the first piece.
How to measure the section
Use calipers or a tape on the actual piece. Measure the thickness in the direction perpendicular to the load as width, and the dimension in line with the load as height. For lumber, remember planing takes a half inch or more off the nominal dimensions. For steel plate or bar, use the mill thickness.
If the piece is notched, drilled, or has a hole cut through it, the real I is lower. This tool handles clean rectangles only.
Common mistakes
- Swapping width and height. This is the most common error, and it can change I by a factor of five or more.
- Using nominal lumber sizes. A 2x10 is not 2 by 10.
- Applying the rectangular formula to a built-up beam. Glued or nailed layers may not act as one solid piece.
- Mixing units. Enter inches so I comes out in in⁴ that other beam formulas expect.
- Treating I alone as a pass or fail result. Strength, deflection, and connections need to be checked by someone qualified.
Related calculators
The beam span calculator uses I to find how long a beam can run before it sags too much. The bending stress calculator turns a moment and a section into psi. Structural framing for houses and decks should be sized by an engineer or against the span tables your local code adopts.
Frequently asked questions
What is the moment of inertia formula for a rectangle?
For a rectangle bending about its centroidal axis, I = width × height³ ÷ 12. A 1.5 in wide by 9.25 in tall section gives 1.5 × 9.25³ ÷ 12 = 98.93 in⁴.
What is section modulus?
Section modulus S is I divided by the distance from the neutral axis to the outer edge. For a rectangle it is width × height² ÷ 6. It lets you find bending stress from a moment: stress equals moment divided by S.
Which side is the height?
Height is the dimension in the direction of the load. A joist on edge has its tall side vertical, so height is the larger number. Lay the same board flat and the height becomes the small number, which drops stiffness sharply.
Do I use nominal or actual lumber sizes?
Use actual dimensions. A 2x10 is about 1.5 by 9.25 in, which is why the calculator starts there. Nominal sizes overstate stiffness.
Does this work for I-beams or round bars?
No, only solid rectangles. I-beams, tubes, and angles have their own formulas or are listed in steel manuals. Use published section properties for those.
Estimates only. Verify quantities with your supplier before purchasing.