Area, centroid, moment of inertia Ix, section modulus Zx and radius of gyration rx for a rectangle, hollow rectangle, circle, T- or I-section.
For this doubly-symmetric section the centroid sits at mid-depth (d = h/2), so Zₓ,top = Zₓ,bot.
Worked examples
All moment of inertia formulas →Need to design this section, not just measure it?
Take these properties into a full code check — capacity, stresses and utilisation.
Reference only. Sections are idealised as plates — fillet radii, root toes and fabrication tolerances of rolled shapes are ignored, so values differ slightly from tabulated properties. The T- and I-sections assume the flange and web are perfectly rectangular. Always verify against the relevant section tables or a full design check.
How the Section Property Calculator works
Every bending, deflection and buckling check starts from the geometry of the cross-section. This calculator takes the dimensions of a rectangle, hollow rectangle, solid circle, T-section or symmetric I-section in millimetres or inches and returns the area A, the centroid height ȳ measured from the bottom, the moment of inertia Ix about the horizontal centroidal axis, the section moduli Zx,top and Zx,bot, and the radius of gyration rx. For the rectangle, hollow rectangle, circle and I-section it also gives Iy and ry.
The rectangle, hollow rectangle, circle and I-section use closed-form expressions. The hollow rectangle and the I-section are treated as an outer rectangle minus the inner void, so Ix = (b·h³ − bi·hi³)/12 for the box and (bf·h³ − (bf − tw)·hw³)/12 for the I, where hw = h − 2tf is the clear web height.
The T-section is built from two rectangles, a web of height h − tf and a flange on top. The centroid is the area-weighted average of the two part centroids, and Ix is found with the parallel-axis theorem: each part adds its own inertia plus its area times the square of the distance from its centroid to the section centroid. Because a T is not symmetric about the x-axis, the top and bottom fibres are at different distances from the centroid, so Zx,top and Zx,bot are reported separately.
Inputs are checked before anything is calculated: dimensions must be positive, the wall of a hollow rectangle must satisfy 2t < min(b, h), the web cannot be wider than the flange, and the flanges must fit in the depth. Sections are idealised as flat plates, so root fillets and toe radii of rolled steel sections are ignored.
- Rectangle
- A = b·h, Ix = b·h³/12, Zx = b·h²/6, rx = h/√12
- Hollow rectangle
- Ix = (b·h³ − bi·hi³)/12, bi = b − 2t, hi = h − 2t
- Solid circle
- A = π·D²/4, Ix = Iy = π·D⁴/64, Zx = π·D³/32, r = D/4
- Symmetric I-section
- A = 2·bf·tf + tw·hw, Ix = (bf·h³ − (bf − tw)·hw³)/12, hw = h − 2tf
- Centroid of a built-up section
- ȳ = Σ(A·y) / ΣA
- Parallel-axis theorem
- Ix = Σ(Io + A·d²), d = y − ȳ
- Section modulus
- Zx,top = Ix / (h − ȳ), Zx,bot = Ix / ȳ
- Radius of gyration
- r = √(I / A)
Worked example: moment of inertia of a flanged (T) beam
Given
- Flange width bf = 1200 mm, flange thickness tf = 120 mm
- Web thickness tw = 300 mm
- Overall depth h = 600 mm
Steps
- Web: height 600 − 120 = 480 mm. Area = 300 × 480 = 144,000 mm², centroid at 480/2 = 240 mm from the bottom, Io = 300 × 480³ / 12 = 2,764,800,000 mm⁴.
- Flange: area = 1200 × 120 = 144,000 mm², centroid at 480 + 120/2 = 540 mm from the bottom, Io = 1200 × 120³ / 12 = 172,800,000 mm⁴.
- Total area A = 144,000 + 144,000 = 288,000 mm².
- Centroid ȳ = (144,000 × 240 + 144,000 × 540) / 288,000 = 390 mm from the bottom.
- Offsets: web d = 240 − 390 = −150 mm, flange d = 540 − 390 = 150 mm. Each A·d² = 144,000 × 150² = 3,240,000,000 mm⁴.
- Ix = (2,764,800,000 + 3,240,000,000) + (172,800,000 + 3,240,000,000) = 9,417,600,000 mm⁴ = 941,760 cm⁴.
- Zx,top = 9,417,600,000 / (600 − 390) = 44,845,714 mm³ and Zx,bot = 9,417,600,000 / 390 = 24,147,692 mm³.
- rx = √(9,417,600,000 / 288,000) = √32,700 = 180.8 mm.
Result: A = 288,000 mm², ȳ = 390 mm, Ix = 9,417,600,000 mm⁴ (941,760 cm⁴), Zx,top = 44,845,714 mm³, Zx,bot = 24,147,692 mm³, rx = 180.8 mm.
Enter the same values in the calculator above to check each step.
Frequently asked questions
How to calculate moment of inertia of a rectangular beam section?
Use Ix = b·h³/12 about the horizontal axis through the centroid, with h being the depth in the direction of bending. A 300 × 500 mm beam gives 300 × 500³ / 12 = 3,125,000,000 mm⁴, which is 312,500 cm⁴. Its section modulus is b·h²/6 = 12,500,000 mm³.
How to find the section modulus of an I-section?
Find Ix as the outer rectangle minus the two voids beside the web, (bf·h³ − (bf − tw)·hw³)/12, then divide by half the depth. For bf = 300, tf = 20, tw = 12 and h = 500 mm, hw = 460 mm and Ix = 788,936,000 mm⁴, so Zx = 788,936,000 / 250 = 3,155,744 mm³. This is the elastic modulus; the plastic modulus Zp is not calculated.
Why are the top and bottom section moduli of a T-section different?
The centroid of a T sits closer to the flange, so the extreme fibres are at different distances from the neutral axis. In the 1200 × 120 flange, 300 × 600 overall T-beam the centroid is 390 mm from the bottom, giving Zx,top = Ix/210 = 44,845,714 mm³ and Zx,bot = Ix/390 = 24,147,692 mm³. The smaller value governs the stress at the bottom fibre.
What is the radius of gyration of a circular section?
For a solid circle r = √(I/A) simplifies to D/4 about any diameter. A 400 mm diameter column has A = 125,663.7 mm², I = π × 400⁴ / 64 = 1,256,637,061 mm⁴ and r = 100 mm.
Will the results match the IS steel tables for rolled sections?
Not exactly. The calculator treats flanges and webs as perfect rectangles and ignores root fillets, toe radii and tapered flanges, so rolled-section values from the tables will differ slightly. It also handles only symmetric I-sections and T-sections, gives no Iy for the T, and does not cover angles, channels or plastic section modulus.