Section Properties Calculator
Pick from 36 sections — rolled and folded profiles, straight-edged shapes, curved outlines and parabolic areas — enter the dimensions it asks for, and this returns the full set of geometric properties a beam calculation needs — area, centroid, second moments about both axes and about the outer edges, polar moments, radii of gyration, and both the elastic and plastic section moduli. Every value is shown beside the formula that produced it, so the working can be checked rather than trusted. These are geometry, not code provisions: the same numbers appear in NDS, AISC, Eurocode or a nineteenth-century textbook.
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Second moment of area, Ix
49.280 × 10^6 mm⁴
Elastic modulus, Sx
492.800 × 10^3 mm³
Details
Updates as you typeEach section reads only the boxes it needs — a circle ignores the height and width, a solid one ignores the hole — so whatever the others hold cannot stop it answering.
Switching converts what you have already typed rather than reinterpreting it.
Regular polygons only.
The taper of a tapered flange, the angle a sector or segment subtends, or the rotation of a rotated rectangle. Sectors usually want a much larger value than the default.
Summary
Second moment of area, Ix
49.280 × 10^6 mm⁴
Bounding envelope
24,000.00 mm²
- Material9,600.00 mm²40%
- Void14,400.00 mm²60%
- Area, A
- 9.600 × 10^3 mm²
- Elastic modulus, Sx
- 492.800 × 10^3 mm³
- Plastic modulus, Zx
- 624.000 × 10^3 mm³
- Radius of gyration, Kx
- 71.6473 mm
- Second moment, Iy
- 19.080 × 10^6 mm⁴
- Shape factor, Zx / Sx
- 1.266
- This section is symmetric about both axes, so the centroid is at mid-depth and both faces reach yield together.
- Subscript 1 marks an axis along the outer edge rather than through the centroid. Design almost always wants the centroidal values; the edge values are what you carry into a further parallel-axis shift.
- The plastic modulus Z is taken about the equal-area axis, which on an asymmetric section is a different line from the centroid. Using the centroid instead would return a larger — and unconservative — value.
How this is calculated
- A = ab − a₁b₁
- 9.600 × 10^3 mm²
- Po — outer perimeter
- 640.0000 mm
- Pi — inner perimeter
- 500.0000 mm
- cx — centroid from the left
- 60.0000 mm
- cy — centroid from the bottom
- 100.0000 mm
- Ix = (ba³ − b₁a₁³) / 12
- 49.280 × 10^6 mm⁴
- Iy = (b³a − b₁³a₁) / 12
- 19.080 × 10^6 mm⁴
- Ix₁ = Ix + A·cy²
- 145.280 × 10^6 mm⁴
- Iy₁ = Iy + A·cx²
- 53.640 × 10^6 mm⁴
- Jz = Ix + Iy
- 68.360 × 10^6 mm⁴
- Jz₁ = Ix₁ + Iy₁
- 198.920 × 10^6 mm⁴
- Kx = √(Ix / A)
- 71.6473 mm
- Ky = √(Iy / A)
- 44.5814 mm
- Kz = √(Jz / A)
- 84.3850 mm
- Kx₁ = √(Ix₁ / A)
- 123.0176 mm
- Ky₁ = √(Iy₁ / A)
- 74.7496 mm
- Kz₁ = √(Jz₁ / A)
- 143.9473 mm
- Sx = Ix / max(cTop, cBot)
- 492.800 × 10^3 mm³
- Sy = Iy / max(cLeft, cRight)
- 318.000 × 10^3 mm³
- Zx = (ba² − b₁a₁²) / 4
- 624.000 × 10^3 mm³
- Zy = (ab² − a₁b₁²) / 4
- 396.000 × 10^3 mm³
Inner height (mm)
Compare scenarios
See how one change moves the result
- CurrentYour inputs as they stand49.280 × 10^6 mm⁴Current
- Outer height, amm 250125.530 × 10^6 mm⁴
- Outer width, bmm 15069.280 × 10^6 mm⁴
Frequently asked questions
What is the second moment of area, and why does it matter more than area?
It measures how the material is spread about the bending axis, and it is what governs both stiffness and bending stress. Because it depends on the cube of the depth, moving material away from the centroid pays enormously: doubling the depth of a rectangle multiplies Ix by eight while only doubling the area. That is the entire reason beams are deep rather than wide, and why a hollow section keeps most of its stiffness after most of its material has gone.
What is the difference between the elastic and plastic section modulus?
The elastic modulus S is Ix divided by the distance to the extreme fibre, and it tells you the moment at which the outermost material first reaches yield. The plastic modulus Z is the first moment of both halves of the area about the centroid, and it tells you the moment at which the whole section has yielded. Z is always the larger, and their ratio is the shape factor — exactly 1.5 for a solid rectangle, and lower for a hollow one because the material that yielding would have recruited is the material you removed.
Why are there two sets of second moments, with and without the subscript?
The unsubscripted values are about axes through the centroid, which is what a design check almost always wants. The subscripted ones are about axes along the outer edges, obtained by the parallel-axis theorem: Ix₁ = Ix + A·cy². Those are useful when this section is one part of a larger built-up shape, because you can shift each part to a common axis and add them.
Does this assume the hole is in the middle?
Yes. The inner rectangle is taken as concentric with the outer one, which is what makes the arithmetic a simple subtraction — the two shapes share a centroid, so their second moments subtract directly. An off-centre hole needs the parallel-axis theorem applied to the hole as well, and would give a centroid that is no longer at half the height.
Are these values code-compliant?
They are geometry, so they are the same in every code — the area and second moment of a rectangle do not change between NDS, AISC, AISI and Eurocode. What differs between codes is what you then do with them: the material strengths, the adjustment factors, the resistance or safety factors, and the slenderness limits. This page gives you the section properties; the code check that follows is a separate step.