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
Related tools
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
- 1Choose the shape: Pick a rectangle, hollow rectangle, circle, hollow circle or pipe, isosceles triangle, I-beam, T-beam or channel, or start from a preset section.
- 2Choose the unit: Select mm, cm, m, in or ft; all dimensions use the same unit and the results come out in that unit to the fourth power.
- 3Enter the dimensions: Type the widths, heights, diameters and thicknesses shown in the diagram for the selected shape.
- 4Read the results: Ix, Iy, the polar moment Ip, the section moduli Sx and Sy, the area and the centroid update as you type, with the formula used.
- 5Copy or export: Copy the results or export them as a text report.
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 — isosceles 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 | 42.67 in⁴ |
| Solid Circle | d=4 in | 12.57 in⁴ | 12.57 in⁴ |
| Hollow Circle | D=5 in, d=3 in | 26.70 in⁴ | 26.70 in⁴ |
| Triangle (isosceles) | b=6 in, h=8 in | 85.33 in⁴ | 36.00 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 |
| Isosceles 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 area moment of inertia?
A measure of how a cross-section's area is spread around an axis, in length⁴ (mm⁴ or in⁴). The larger it is, the stiffer the beam in bending: deflection is inversely proportional to E × I. For a rectangle, I = bh³/12, so doubling the depth makes a beam eight times stiffer.
What is the difference between Ix and Iy?
Ix is taken about the horizontal axis through the centroid and governs bending under vertical loads, the usual case for a floor beam. Iy is about the vertical axis and governs sideways bending and buckling about the weak axis. For a circle or square they are equal.
What is the section modulus?
S = I ÷ c, where c is the distance from the centroid to the farthest edge. The bending stress is σ = M ÷ S. For shapes that are not symmetric, such as a T-beam, a channel or a triangle, the calculator uses the farthest edge, which gives the largest stress.
What is the polar moment of inertia?
Ip = Ix + Iy, the resistance of a section to twisting about its centroid. For solid and hollow circular shafts it equals the torsion constant J used in τ = T·r ÷ J. For open sections such as I-beams and channels the real torsion constant is much smaller, so do not use Ip for their twist.
Why are I-beams so efficient?
Most of their material sits in the flanges, far from the neutral axis, and moment of inertia grows with the square of that distance. An I-beam therefore gives a much higher Ix than a solid rectangle of the same weight, while the thin web carries the shear.
Is this the same as mass moment of inertia?
No. The area moment of inertia (length⁴) describes a cross-section's resistance to bending. The mass moment of inertia (kg·m² or lb·ft²) describes a body's resistance to angular acceleration and is used for flywheels, rotors and anything that spins.
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