Section Properties & Moment of Inertia
Area, centroid, second moment of area, section modulus and radius of gyration for rectangles, hollow sections, circles, tubes, I-beams, tees, channels, angles.
A, centroid position, Ixx and Iyy about the centroid, Zxx and Zyy (both fibres for asymmetric shapes), rx and ry and the perimeter — pure geometry with the formula used.
Example: An I-section 100 × 200 with 6 mm web and 10 mm flanges: A 3,080 mm², Ixx 2.098 × 10⁷ mm⁴, Zxx 209,827 mm³, rx 82.5 mm; a 100 × 100 tee has its centroid 71.3 mm from the bottom.
Rectangles,
parallel axes.
How the shapes are built, what each property means, and where catalogue sections differ.
Building the shape
Every straight-sided section is a set of rectangles: an I-beam is two flanges and a web, a tee a flange and a stem, a channel a web and two flanges, an angle two legs. The area is their sum and the centroid the area-weighted mean of their centres; the second moment of area about the centroid is Σ(bh³/12 + A·d²) — each rectangle’s own term plus the parallel-axis shift. Circles and tubes use π(D⁴ − d⁴)/64.
The properties
I (second moment of area) measures bending stiffness; Z = I ÷ c, the elastic section modulus, gives the bending stress for a moment (σ = M/Z) at the fibre farthest from the axis — for a tee or angle the near fibre’s Z is shown too, since the two differ; r = √(I/A) is the radius of gyration used for slenderness.
Limits
Pure geometry: no material, load or safety factor. Rolled sections have root fillets and sometimes tapered flanges that add a little area and I — use the maker’s table for a catalogue section. An angle’s principal axes are rotated from the leg-parallel axes used here, so its true minimum I is smaller. Nothing leaves the browser; the same four anonymous usage counts as the rest of the site apply.
SOURCES
- Composite rectangles with the parallel-axis theorem: I = Σ(bh³/12 + A·d²); circles π(D⁴ − d⁴)/64; Z = I ÷ c; r = √(I/A) — mechanics of materials; catalogue sections with fillets differ slightly
Last reviewed 21 September 2026. How results are checked: How we verify.