Calculator Inputs
Use the form below to calculate area, centroidal moment of inertia, and section modulus for several common shapes.
Example Data Table
The sample values below use millimeters for dimensions. Outputs are shown in mm², mm³, and mm⁴.
| Shape | Dimensions | Area | Ix | Iy | Zx Governing | Zy Governing |
|---|---|---|---|---|---|---|
| Solid Rectangle | b = 120, h = 240 | 28,800.0000 | 138,240,000.0000 | 34,560,000.0000 | 1,152,000.0000 | 576,000.0000 |
| Solid Circle | D = 100 | 7,853.9816 | 4,908,738.5212 | 4,908,738.5212 | 98,174.7704 | 98,174.7704 |
| Isosceles Triangle | b = 180, h = 240 | 21,600.0000 | 69,120,000.0000 | 29,160,000.0000 | 432,000.0000 | 324,000.0000 |
| Hollow Circle | D = 160, d = 100 | 12,252.2113 | 27,261,170.2515 | 27,261,170.2515 | 340,764.6281 | 340,764.6281 |
| Ellipse | w = 160, h = 100 | 12,566.3706 | 7,853,981.6340 | 20,106,192.9830 | 157,079.6327 | 251,327.4123 |
Formula Used
General Relations
Z = I / c
Section modulus equals moment of inertia divided by the distance from the neutral axis to the extreme fiber.
Solid Rectangle
A = b × h
Ix = b × h³ / 12
Iy = h × b³ / 12
Zx = Ix / (h/2), Zy = Iy / (b/2)
Hollow Rectangle
A = B × H − b × h
Ix = (B × H³ − b × h³) / 12
Iy = (H × B³ − h × b³) / 12
Zx = Ix / (H/2), Zy = Iy / (B/2)
Solid Circle
A = π × D² / 4
Ix = Iy = π × D⁴ / 64
Zx = Zy = I / (D/2)
Hollow Circle
A = π × (D² − d²) / 4
Ix = Iy = π × (D⁴ − d⁴) / 64
Zx = Zy = I / (D/2)
Isosceles Triangle
A = b × h / 2
Ix = b × h³ / 36
Iy = b³ × h / 48
Zx(top) = Ix / (2h/3), Zx(bottom) = Ix / (h/3), Zy = Iy / (b/2)
Ellipse
A = π × w × h / 4
Ix = π × w × h³ / 64
Iy = π × h × w³ / 64
Zx = Ix / (h/2), Zy = Iy / (w/2)
How to Use This Calculator
- Choose the section shape that matches your beam, bar, or profile.
- Select a consistent unit system such as mm, cm, m, in, or ft.
- Enter the required dimensions for the selected shape.
- Click Calculate Now to display area, Ix, Iy, and governing section modulus values.
- Review the graph, compare axis behavior, and export the results as CSV or PDF.
- Use the example table and formulas to verify your entries or explain the result to others.
FAQs
1. What does moment of inertia represent?
Moment of inertia describes how strongly a section resists bending about a chosen axis. Larger values usually mean better bending stiffness for that axis.
2. What is section modulus used for?
Section modulus connects geometry to bending stress. Engineers use it with bending moment to estimate elastic stress at the outermost fibers.
3. Why are Ix and Iy different?
They differ because the section shape is not equally distributed about both axes. A tall section boosts Ix, while a wide section boosts Iy.
4. Why does the triangle show two Zx values?
The centroid of a triangle is not halfway up the height. That makes the distance to the top and bottom fibers different, so top and bottom section modulus values are not equal.
5. Can I use any unit system?
Yes. Enter all dimensions in one consistent unit system. The calculator automatically reports area, inertia, and modulus in matching derived units.
6. Are hollow sections always more efficient?
Often, yes. Hollow sections can place more material away from the neutral axis, which improves inertia and stiffness for the same general amount of material.
7. Does this calculator include material strength?
No. It evaluates geometric properties only. You must combine these results with material strength, loading, and design rules for full beam design.
8. What should I check before trusting the result?
Confirm the shape selection, enter dimensions in a single unit system, and make sure inner dimensions are smaller than outer dimensions for hollow sections.