BIZOMAT MAKEOS · Strength
Section Properties: Ellipse, Hexagon, Triangle, Z and Cross
What are the area, second moments and section moduli of this section?
Area, centroid, second moments of area, section moduli, radii of gyration and polar moment of elliptical, hollow elliptical, half-round, hexagonal, triangular, round-with-hexagon-hole, round-with-square-hole, Z and cross (X) sections.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Section | choice | Section type; the option text names each dimension. |
| Dimension A | mm | First dimension as described. |
| Dimension B | mm | Second dimension where needed. |
| Dimension C | mm | Third dimension where needed. |
| Dimension D | mm | Fourth dimension where needed. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Section
- What it is
- The cross-section shape whose properties you want. Each option names what Dimensions A to D mean for that shape.
- Units and format
- Choose one of 9 options from the list: Ellipse (A, B = full axes); Hollow ellipse (A, B outer; C, D inner axes); Half circle (A = diameter); Hexagon (A = across flats); Isosceles triangle (A = base, B = height); Round with hexagonal hole (A = dia, C = hole across flats); Round with square hole (A = dia, C = hole side); Z section (A = depth, B = flange width, C = flange t, D = web t); Cross / X section (A = height, B = width, C = thickness of both bars).
- Example
- The form opens with “Hexagon (A = across flats)” selected as a worked example. Choose the option that matches your case.
- Get it right
- Ellipse axes are full widths, not semi-axes; a hexagon is sized across flats, not across corners.
What this means: Dimension A
- What it is
- The main size of the section: the horizontal full axis of an ellipse, the diameter of a half circle or round section, the across-flats size of a hexagon, the base of a triangle, or the overall depth or height of a Z or cross section.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Check the option text: for an ellipse A is a full axis, not a semi-axis.
What this means: Dimension B
- What it is
- The second size where the shape has one: the vertical full axis of an ellipse, the height of a triangle, the flange width of a Z section or the overall width of a cross section.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Shapes that do not name B ignore it.
What this means: Dimension C
- What it is
- The third size where the shape has one: the inner horizontal axis of a hollow ellipse, the hole size in a round section, the flange thickness of a Z section or the bar thickness of a cross section.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A hexagonal hole is sized across flats; a square hole by its side.
What this means: Dimension D
- What it is
- The fourth size, used only by the hollow ellipse (its inner vertical axis) and the Z section (its web thickness).
- Units and format
- Enter a number in mm: millimetres. The form accepts 0 mm to 100000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 5 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Other shapes ignore it. For a hollow ellipse it must be smaller than B.
Method and formulas
- Ellipse: I = πab³/4 about the major axis (a, b semi-axes); hollow ellipse subtracts the inner ellipse.
- Half circle: centroid 4r/3π; Ix = (π/8 − 8/9π) r⁴. Hexagon (side s): I = 5√3 s⁴/16 about any centroidal axis. Triangle: Ix = bh³/36.
- Round with a hole: subtract the hole’s properties. Z and cross sections: sum of rectangles with the parallel axis theorem; principal moments for the Z.
- Example: a 30 mm across-flats hexagon has A = 779.42 mm² and I = 48,714 mm⁴.
Limitations
- Sharp corners; rolled and drawn sections have radii that change the properties slightly.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find properties of bars, extrusions and fabricated sections
- Check torsion and bending of hex and special bars
- Replace handbook section tables
Who it is for
- Design engineers
- Structural and mechanical engineers
- Engineering students
What you get
- Area and centroid
- Ix and Iy about centroidal axes
- Section moduli and radii of gyration
- Polar moment where meaningful
Interpreting the result
- For torsion of non-circular sections, use the torsion tools rather than the polar moment.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 234–246. Topic reference; no source tables or recommended settings are reproduced.
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Open the Section Properties: Ellipse, Hexagon, Triangle, Z and Cross calculator