Productive ToolboxMoment of Inertia Calculator
Calculate area moment of inertia (Ix, Iy, polar moment) for rectangles, circles, hollow sections, I-beams, T-beams, and more. Instant engineering-grade results.
Moment of Inertia Calculator
Calculate area moment of inertia (Ix, Iy), polar moment, and section modulus for common structural cross-sections. Used in beam bending, shaft design, and structural analysis.
Moment of Inertia
Ix (about x-axis)
Settings & Actions
Shape & Unit
💡 Used commonly in beam and column design.
Dimensions
Cross-Section Diagram
Quick Presets
What is the Moment of Inertia?
The Area Moment of Inertia (also called the Second Moment of Area) is a geometric property of a cross-section that measures its resistance to bending. It is one of the most fundamental values in structural and mechanical engineering, used in beam deflection, bending stress, and shaft design calculations.
A larger moment of inertia means the cross-section is stiffer and resists bending more effectively. This is why I-beams are used in construction — their shape maximizes the moment of inertia while minimizing material use.
This calculator computes Ix (about the x-axis), Iy (about the y-axis), the polar moment Ip, and the section modulusfor rectangles, circles, hollow sections, I-beams, T-beams, channel sections, and pipes.
How to Use This Calculator
Step-by-Step
- 1Select the cross-section shape from the dropdown
- 2Choose your unit (mm, cm, m, in, or ft)
- 3Enter the dimensions for the selected shape
- 4Results update instantly — Ix, Iy, Ip, Sx, Sy, Area, and Centroid
- 5Use presets for common structural sections
- 6Export the result as a TXT report or copy to clipboard
Supported Shapes
- ✓Rectangle — solid rectangular cross-section
- ✓Hollow Rectangle — box beam section
- ✓Solid Circle — round bar or rod
- ✓Hollow Circle — pipe or hollow shaft
- ✓Triangle — triangular cross-section
- ✓I-Beam — standard structural steel section
- ✓T-Beam — reinforced concrete T-section
- ✓Channel Section — C-channel structural member
- ✓Pipe Section — circular hollow section
Example Calculations
| Shape | Dimensions | Ix | Iy |
|---|---|---|---|
| Rectangle | b=4 in, h=8 in | 170.67 in⁴ | 21.33 in⁴ |
| Solid Circle | d=4 in | 12.57 in⁴ | 12.57 in⁴ |
| Hollow Circle | D=5 in, d=3 in | 42.88 in⁴ | 42.88 in⁴ |
| Triangle | b=6 in, h=8 in | 85.33 in⁴ | 27.00 in⁴ |
| Rectangle | b=200 mm, h=400 mm | 1,066,666,666.67 mm⁴ | 266,666,666.67 mm⁴ |
| Solid Circle | d=100 mm | 4,908,738.52 mm⁴ | 4,908,738.52 mm⁴ |
Formula Reference
| Shape | Ix Formula | Iy Formula |
|---|---|---|
| Rectangle | bh³ / 12 | hb³ / 12 |
| Hollow Rect | (bh³ − b_i·h_i³) / 12 | (hb³ − h_i·b_i³) / 12 |
| Solid Circle | πd⁴ / 64 | πd⁴ / 64 |
| Hollow Circle | π(D⁴ − d⁴) / 64 | π(D⁴ − d⁴) / 64 |
| Triangle | bh³ / 36 | hb³ / 48 |
| I-Beam | (bf·H³ − (bf−tw)·hw³) / 12 | (2tf·bf³ + hw·tw³) / 12 |
Who Uses This Calculator?
Structural Engineers
Analyze beam bending, deflection, and load capacity for structural members.
Mechanical Engineers
Design shafts, axles, and machine components under bending and torsional loads.
Engineering Students
Verify textbook calculations and understand cross-section properties.
Civil Engineers
Size beams, columns, and slabs for buildings and infrastructure.
CAD Designers
Validate cross-section properties during component design.
Manufacturing
Select standard sections for fabricated structural components.
Frequently Asked Questions
What is the difference between Ix and Iy?
Ix is the moment of inertia about the horizontal (x) axis, resisting vertical bending. Iy is about the vertical (y) axis, resisting horizontal bending. For symmetric shapes like circles, Ix = Iy.
What is the polar moment of inertia?
The polar moment of inertia (Ip = Ix + Iy) measures resistance to torsion (twisting). It is used in shaft design to calculate shear stress under torque.
What is the section modulus?
Section modulus S = I / c, where c is the distance from the neutral axis to the extreme fiber. It is used to calculate bending stress: σ = M / S. A higher section modulus means lower bending stress for the same moment.
Why does the I-beam have a high moment of inertia?
The I-beam concentrates material far from the neutral axis (in the flanges), which maximizes the moment of inertia for a given cross-sectional area. This makes it very efficient for resisting bending.
Is this calculator accurate for engineering use?
Yes. All formulas are based on standard engineering mechanics. Results are computed using double-precision floating point arithmetic. For critical structural applications, always verify with a licensed engineer.
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