Section Properties Calculator
Area, centroid, moment of inertia, section modulus, plastic modulus and radius of gyration for rectangles, tubes, I-beams, tees, channels and angles.
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About this toolWhat is the Section Properties Calculator?
Bending stress, deflection and buckling all depend on numbers that come from the shape of the cross-section alone: the area A, the position of the centroid, the second moment of area I (often called the area moment of inertia), the elastic section modulus S, the plastic modulus Z and the radius of gyration r. This section properties calculator works them out for eight common shapes and draws the section with its centroidal axes so you can see what you described.
Shapes include a solid rectangle, a hollow rectangle or square tube, a solid circle, a round tube or pipe, a doubly symmetric I-beam or H-section, a T-section, a channel and an angle. For unsymmetrical tees, channels and angles you get the section modulus at both extreme fibres, because they differ, and for angles the principal moments of inertia and the minimum radius of gyration that controls their buckling.
Pick millimetres, centimetres, metres or inches. Every result is shown in your unit and a second one (mm and cm, or inches and mm), so you can compare directly with a steel table without converting powers of ten by hand.
- Eight shapes: rectangle, hollow rectangle, circle, tube, I-beam, T, channel and angle
- Area, centroid, Ix, Iy, Sx and Sy at each fibre, Zx, Zy, rx and ry
- Product of inertia, principal moments and minimum r for angles; polar moment J for circles and tubes
- Scaled cross-section drawing with centroid and axes
- Results in two units side by side, copy summary and share link
How to use the Section Properties Calculator
- Pick the shape and the unit you want to work in.
- Enter the dimensions shown for that shape. Flange widths, depths and thicknesses are outside dimensions; for tubes enter the outside diameter or size and the wall thickness.
- Check the drawing: the red dashed line is the x-x centroidal axis and the green one is y-y.
- Read the headline area, Ix, minimum Sx and Zx, then the full table with both units.
- For angles, use the principal values and the minimum radius of gyration for buckling checks. Copy the table or share the link.
Worked examples
Built-up I-beam (default)
Input
bf = 200, d = 300, tf = 15, tw = 10 mm
Output
A = 8,700 mm²; Ix = 138.35 x 10⁶ mm⁴ (13,835 cm⁴); Sx = 922,350 mm³; Zx = 1,037,250 mm³
The shape factor Zx/Sx is 1.12, typical of an I-section: most of the material is already in the flanges, so there is little extra capacity after first yield.
Timber joist, 89 x 235 mm
Input
Rectangle b = 89 mm, h = 235 mm
Output
Ix = 96.25 x 10⁶ mm⁴; Iy = 13.81 x 10⁶ mm⁴; Sx = 819,170 mm³
On edge the joist is about seven times stiffer than laid flat, which is (235/89)² = 7.0.
T-section with unequal fibres
Input
bf = 150, d = 150, tf = 12, tw = 8 mm
Output
ȳ = 115.5 mm from the bottom; Sx,top = 162,920 mm³; Sx,bot = 48,690 mm³
The stem tip is more than three times as far from the centroid as the flange top, so the stem sees more than three times the stress. Design with the smaller S.
Unequal angle 100 x 75 x 8
Input
Horizontal leg 100, vertical leg 75, t = 8 mm
Output
A = 1,336 mm²; Ix = 0.656 x 10⁶ mm⁴; Iy = 1.349 x 10⁶ mm⁴; minimum r = 16.2 mm
The minimum radius of gyration, about an inclined principal axis, is well below rx and ry, and that is the one that governs a single-angle strut.
Background and insight
The calculator splits each shape into rectangles, finds the centroid from the first moment of area, ȳ = ΣAy/ΣA, and then applies the parallel axis theorem, I = Σ(bh³/12 + A d²), to move each piece's own inertia to the common centroidal axis. Hollow shapes subtract the hole. Circles use the exact π D⁴/64. The product of inertia Ixy is computed the same way, and for angles Mohr's circle gives the principal values and the rotation of the principal axes.
The elastic section modulus S = I/c, with c the distance from the centroid to the extreme fibre, converts a bending moment into the maximum elastic stress, σ = M/S. The plastic modulus Z is the first moment of area about the plastic neutral axis, which divides the area into two equal halves; M = Fy Z is the moment at which the whole section has yielded. The ratio Z/S, the shape factor, is 1.5 for a rectangle, 1.70 for a solid circle and roughly 1.1 to 1.2 for I-beams bent about the strong axis. AISC calls these S and Z; Eurocode 3 calls them Wel and Wpl.
Rolled sections have root fillets between web and flange and rounded toes, which this tool ignores. The effect is small but real: an L4x4x1/2 angle calculated here has A = 3.75 in² and ȳ = 1.18 in, matching the AISC Manual, while Ix comes out at 5.56 in⁴ against the tabulated 5.52 in⁴, less than 1% apart.
Common use cases
Beam and column checks
Get I for deflection, S for bending stress and r for slenderness when a section is not in a published table.
Fabricated and built-up sections
Work out properties of plate girders, welded tees and box sections cut from plate.
Timber and aluminium extrusions
Find the stiffness of joists, posts and simple extrusion profiles from their dimensions.
Coursework
Check centroid and parallel-axis calculations for mechanics of materials problems.
Tips and limitations
- Moment of inertia scales with the fourth power of size and S with the cube, so unit mistakes are huge: 1 cm⁴ is 10,000 mm⁴, and 1 in⁴ is 41.62 cm⁴.
- Keep thicknesses realistic. A web or wall thicker than half the section is rejected because the shape no longer exists.
- For tees and channels, the section modulus that matters is the smaller one, at the fibre farthest from the centroid.
- The plastic modulus here assumes one material throughout. Composite sections need a transformed area first.
- Use published section tables for rolled shapes in final design, because they include fillets and are what the code values are based on.
Frequently asked questions
What is the moment of inertia of a rectangle?
About its centroidal axis parallel to the width, I = bh³/12, where b is the width and h the depth. A 100 x 200 mm rectangle has Ix = 66.7 x 10⁶ mm⁴ on edge and only 16.7 x 10⁶ mm⁴ laid flat.
What is the difference between section modulus and moment of inertia?
The moment of inertia I measures stiffness and is used for deflection and buckling. The section modulus S = I/c divides I by the distance to the outermost fibre and converts a bending moment directly into the maximum bending stress, σ = M/S.
What is the plastic section modulus?
Z is the first moment of area of the two halves of the section about the axis that splits the area equally. Multiplying it by the yield stress gives the plastic moment, the full bending capacity of a compact steel section used in AISC LRFD and Eurocode 3 class 1 and 2 design.
How do you find the centroid of a T-section?
Split it into the flange and the stem, multiply each area by the height of its own centroid above the base, add those and divide by the total area. For a 150 x 150 tee with a 12 mm flange and 8 mm stem the centroid is 115.5 mm above the stem tip.
What is the radius of gyration used for?
r = √(I/A) is the distance from the axis at which the whole area could be concentrated to give the same I. It is the denominator of the column slenderness ratio KL/r, so a larger r means a column that resists buckling better for the same weight.
Why do angle sections have principal axes?
An angle is not symmetric about its leg directions, so its product of inertia is not zero. The axes of maximum and minimum I are rotated, at 45 degrees for an equal angle. Buckling and unrestrained bending happen about the minor principal axis.
How do I convert moment of inertia from cm⁴ to mm⁴ or in⁴?
Multiply cm⁴ by 10,000 to get mm⁴, because each centimetre is 10 mm and there are four length dimensions. One in⁴ equals 41.623 cm⁴, or 416,231 mm⁴.
Is there an in-depth guide to the Section Properties Calculator?
Yes. “Moment of Inertia, Section Modulus and Radius of Gyration for Engineers” covers the background, how it works, common pitfalls and worked examples. Read the full guide.
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