How to use it
- Choose the Section shape that matches your beam or shaft.
- Type the dimensions in millimetres into the boxes that the shape shows. For a solid rectangle, the default Width b, or flange width B 50 and Depth h, or overall depth H 100 give a result at once.
- Read Second moment of area I and Section modulus S. For an I-beam, also read the I-beam weak axis I and I-beam weak axis S.
- Open Working, step by step to check each line by hand, with the numbers you typed.
The second moment of area
The second moment of area I measures how the material of a section is spread away from its centroidal axis. The farther the material lies from that axis, the more the section resists bending. Each slice counts with the square of its distance from the axis, so the result has the unit of length to the fourth power, mm⁴ here. Holes are subtracted as negative areas, and an I-section is a rectangle with two voids beside the web.
Rectangle: I = b × h³ ÷ 12
Circle: I = π × d⁴ ÷ 64
Hollow circle: I = π × (D⁴ − d⁴) ÷ 64
Hollow rectangle: I = (B × H³ − b × h³) ÷ 12
I-beam, strong axis: I = (B × H³ − (B − tw) × (H − 2 × tf)³) ÷ 12
I-beam, weak axis: I = (2 × tf × B³ + (H − 2 × tf) × tw³) ÷ 12The section modulus
The section modulus S divides I by the distance c from the centroid to the outermost fibre. For a symmetric section c is half the depth, so a rectangle of depth h has S = b × h² ÷ 6. The calculator gives S next to I because a beam design uses the section modulus together with the bending moment.
S = I ÷ c
c = h ÷ 2 for a rectangle or an I-beam, D ÷ 2 for a circle, D ÷ 2 for the solid circleWhat this calculator does not include
- Rolled steel profiles. The Steel Construction Manual of the AISC lists the area, depth, I and S of the standard steel shapes it covers, and the manufacturer's tables give the exact values of other profiles to use in a design.
- Sections with rounded corners or fillets. The formulas here are for the idealised shapes with sharp corners.
- Stresses, loads and deflections. The calculator gives the properties of the section only, not whether a beam carries a given load.
Worked example: I-beam 100 mm wide and 200 mm deep
An I-beam with a flange width B of 100 mm, an overall depth H of 200 mm, a flange thickness tf of 8 mm and a web thickness tw of 6 mm has an area of 2 × 100 × 8 + (200 − 16) × 6 = 2,704 mm². Its second moment about the strong axis is Ix = (100 × 200³ − 94 × 184³) ÷ 12 = 1.787 × 10⁷ mm⁴, and its section modulus is Sx = Ix ÷ 100 = 1.787 × 10⁵ mm³.
About the weak axis the same beam has Iy = 1.337 × 10⁶ mm⁴ and Sy = 2.673 × 10⁴ mm³.
| Figure | Value |
|---|---|
| Area A | 2,704 mm² |
| Second moment of area, strong axis Ix | 1.787 × 10⁷ mm⁴ |
| Section modulus, strong axis Sx | 1.787 × 10⁵ mm³ |
| Second moment of area, weak axis Iy | 1.337 × 10⁶ mm⁴ |
| Section modulus, weak axis Sy | 2.673 × 10⁴ mm³ |
Frequently asked questions
What is the difference between the moment of inertia and the second moment of area?
For a flat section they are the same quantity: the second moment of area I is the moment of inertia of an area. The mass moment of inertia of a solid body uses its density as well, so it gives a different number.
Why does the depth come in with the third power?
Each slice of the section counts with the square of its distance from the centroid, and summed over a rectangle that gives b × h³ ÷ 12. Doubling the depth therefore multiplies I by eight, while doubling the width only doubles it.
Why does the calculator refuse a hollow shape whose hole is as large as the outside?
A hole that is not smaller than the outside would leave no material or a negative area, and that is not a section. Check that the inner width and depth, or the inner diameter, are each smaller than the outer size.
Is the I-beam value given about the strong axis?
Yes. The first figure is about the horizontal axis through the centroid, which is the strong axis. The weak axis is the vertical one, and its figures are given in the two lines under the main result.
Can I use this for a standard steel beam such as an IPE or an HEA?
Use it for a check, but take the exact values from the manufacturer's table for that profile. For the American shapes the Steel Construction Manual of the AISC lists the area, depth, I and S.
Are my numbers sent to a server?
No. The calculation runs in your browser, and nothing you type is stored or sent to us.
Sources
- Structural Design (open textbook), chapter 8: Cross-sectional Properties, I = bh³/12 and S = I/c
- University of Alberta Engineering Courses, Statics, section 10.4: holes as negative areas and the parallel axis theorem
- IIT Madras NPTEL, Strength of Materials (Prof. M. S. Sivakumar), module 3: polar moment J = π d⁴ ÷ 32 and the tube
Figures and text last checked on Thursday, November 12, 2026.