Moment of Inertia Calculator
Calculate the moment of inertia and section modulus of a rectangular beam cross-section from its width and height.
How this calculator works
Every beam’s resistance to bending comes partly from its material and partly from the shape of its cross-section. Moment of inertia (I) and section modulus (S) capture that shape contribution. This calculator takes a rectangular cross-section’s width and height and returns both values, which then plug into deflection, buckling, and bending-stress formulas elsewhere.
For a rectangle, both values come directly from width and height — no material properties are needed at this step, since I and S describe geometry only.
Formula: Moment of inertia I = width × height³ ÷ 12. Section modulus S = width × height² ÷ 6 (this is I divided by the distance from the neutral axis to the outer fiber, c = height/2).
Worked example
A 1.5 in × 9.25 in section (the actual size of a nominal 2×10):
- I = 1.5 × 9.25³ ÷ 12 ≈ 98.93 in⁴
- S = 1.5 × 9.25² ÷ 6 ≈ 21.39 in³
Notes
I measures a section’s resistance to bending deflection, and it’s used together with the modulus of elasticity (E) in the beam deflection and column buckling formulas. S measures a section’s resistance to bending stress, and it’s used in the stress formula σ = M/S.
Because height is cubed in the I formula, a deeper section gains stiffness much faster than a wider one. Doubling a section’s depth increases its moment of inertia eightfold, while doubling its width only doubles it — which is why going deeper is usually a more efficient way to stiffen a beam than going wider.
How to use
Enter the actual width and height of your rectangular cross-section, in inches — not the nominal lumber size. The calculator returns moment of inertia (I) and section modulus (S), both of which you can carry into the Beam Span Calculator, Bending Stress Calculator, or Beam Deflection Calculator.
This is a planning estimate using standard mechanics formulas, not a substitute for engineering design. Confirm actual member sizing and code compliance with a licensed structural engineer.
Frequently asked questions
What is moment of inertia in beam design?
Moment of inertia (I) is a measure of how a cross-section's shape resists bending. It depends only on geometry — width and height — not material. A deeper or wider section has a higher I, meaning it bends less under the same load. It's used together with the material's modulus of elasticity (E) in deflection and buckling calculations.
What is section modulus and how is it different from moment of inertia?
Section modulus (S) measures a cross-section's resistance to bending stress, while moment of inertia (I) measures its resistance to bending deflection. S is I divided by the distance from the neutral axis to the outer fiber. S is used with material yield strength to check bending strength; I is used with modulus of elasticity to check stiffness.
Why does beam depth matter more than beam width?
In the rectangular formula, height is cubed while width is only multiplied. That means increasing depth has a far bigger effect on stiffness than increasing width by the same amount. Doubling a section's depth increases its moment of inertia eightfold, while doubling its width only doubles it.
Should I use nominal or actual lumber dimensions?
Always use actual (dressed) dimensions, not nominal ones. A nominal 2x10 actually measures 1.5 in by 9.25 in after milling, and that's the size that determines its real moment of inertia and section modulus. Using nominal sizes will overstate a beam's stiffness and strength.
Estimates only. Verify quantities with your supplier before purchasing.