Rectangular section
Given: b = 100 mm, h = 200 mm
Working: I = 100 × 200³ / 12
I = 66,666,666.7 mm⁴.
Structural & Mechanical Engineering
Moment of Inertia Formula
Learn area moment of inertia formulas for rectangles and circles, the parallel-axis theorem and how to calculate section inertia for beam bending and deflection.
Governing equation
Rectangle about centroidal x-axis: I = b × h³ / 12
In beam and structural calculations, the area moment of inertia (second moment of area) describes how cross-sectional area is distributed about an axis. A larger I generally means greater bending stiffness for the same material.
For a solid circle about a centroidal diameter, I = πd⁴/64. To shift a centroidal inertia to a parallel axis, use I = Ic + A d².
Live calculator
Rectangle area moment of inertia calculator
Centroidal x-axis formula I = bh³/12.
Result
66,666,666.7mm⁴
Rectangle about centroidal axis parallel to width
Formula variables and units
| Symbol | Meaning | Typical SI unit |
|---|---|---|
| I | Area moment of inertia | mm⁴ or m⁴ |
| b | Rectangle width | mm or m |
| h | Rectangle depth perpendicular to the axis | mm or m |
| A | Cross-sectional area for parallel-axis theorem | mm² or m² |
How to calculate it
- 1Choose the axis about which bending or stiffness is being evaluated.
- 2Split complex sections into simple shapes when necessary.
- 3Calculate each shape's centroidal inertia.
- 4Use the parallel-axis theorem when a shape centroid does not lie on the target neutral axis.
- 5Sum the contributions using consistent units.
Common mistakes
- Confusing mass moment of inertia with area moment of inertia.
- Using the wrong centroidal axis.
- Forgetting that section depth is raised to the third power for a rectangle.
- Mixing mm⁴ and m⁴.
Worked examples
Orientation effect
Given: Rotate the same 100 × 200 mm rectangle by 90°
Working: I now uses h = 100 mm and b = 200 mm
The inertia about that axis becomes much smaller because depth is cubed.
Go deeper
Use the dedicated OneCalcApp tool
The quick calculator above demonstrates the governing equation. Open the related engineering tool for broader inputs, outputs and calculation context where available.
Frequently asked questions
What is the moment of inertia formula for a rectangle?
About its centroidal axis parallel to width b, the area moment of inertia is I = bh³/12.
What is the moment of inertia formula for a solid circle?
About a centroidal diameter, I = πd⁴/64.
What is the parallel-axis theorem?
It shifts a centroidal second moment of area to a parallel axis using I = Ic + Ad².
Is area moment of inertia the same as mass moment of inertia?
No. Area moment of inertia is a geometric section property used in bending and deflection; mass moment of inertia describes rotational resistance of mass.
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