Area centroid and first moment of area
The area centroid is the area-weighted mean of component centroids; the first moment about an axis is area times perpendicular centroid distance.
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Formula
A in m2; x̄, ȳ, xi, yi in m; Qx and Qy in m3
Variables & units
| Symbol | Meaning | Unit |
|---|---|---|
| Area of disjoint component i | m2 | |
| Coordinates of component i centroid | m | |
| x̄,ȳ | Coordinates of the composite-area centroid | m |
| First area moment about x-axis, defined as integral of y over A | m3 | |
| First area moment about y-axis, defined as integral of x over A | m3 |
LaTeX
\bar{x}=\frac{\sum_i A_i x_i}{\sum_i A_i},\quad \bar{y}=\frac{\sum_i A_i y_i}{\sum_i A_i},\quad Q_x=\int_A y\,dA=A\bar{y}
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Derivation & background
For a composite planar section, add nonoverlapping component areas with signs; holes are negative areas. The centroid follows from the vanishing first moment about its centroidal axes. With the stated convention, Qx=integral y dA=Aȳ and Qy=integral x dA=A x̄. The first moment has units m3 and is not the second moment of area I, which has units m4. The parallel-axis theorem transfers a second moment between parallel axes; it does not locate the centroid, though centroid location is needed before using that theorem.
Exam blueprint
Validity range
Planar area geometry; disjoint components in one coordinate system. Holes have negative area. Qx=∫y dA has m3; I=∫y2dA has m4.
Derivation steps
The centroid is defined so first area moments about its axes vanish.
- 1List component areas Ai and centroid coordinates (xi,yi).
- 2Form total area A=ΣAi and moments Qx=ΣAiyi, Qy=ΣAixi.
- 3Compute ȳ=Qx/A and x̄=Qy/A; use negative areas for holes.
Rearrangements
First moment from centroid
State axis and sign convention.
Composite centroid
All coordinates must share one origin.
Task variant
A T-section base is 60×20 mm with y1=10 mm; its centered web is 20×40 mm with y2=40 mm. Find A, Qx about the bottom edge and ȳ.
A1=1200 mm2, A2=800 mm2, A=2000 mm2.
Qx=1200·10+800·40=44,000 mm3 and ȳ=Qx/A=22 mm.
Check: A ȳ=44,000 mm3.
Common mistakes
Confuse Q with second moment I.
Q has m3 and measures area position; I has m4 and describes bending resistance.
Exam context
- Composite sections and beam-shear calculations.
Formula cluster
Cross-section geometry
First moment locates the centroid; second moment describes bending stiffness.
Worked example
A nonoverlapping T section has a 60×20 mm base, A1=1200 mm2 and y1=10 mm, plus a centered 20×40 mm web above it, A2=800 mm2 and y2=40 mm. Symmetry gives x̄=30 mm. About the bottom edge, Qx=1200·10+800·40=44000 mm3 and A=2000 mm2, so ȳ=22 mm. Check: Aȳ=2000·22=44000 mm3; the centroid lies within the 20–60 mm section height.
Applications
Locate centroids and neutral axes of composite beam sections, and compute first moments used in beam-shear formulas.
How do you calculate with Area centroid and first moment of area?
Here is how to work through a typical Area centroid and first moment of area (x̄=ΣAᵢxᵢ/A; ȳ=ΣAᵢyᵢ/A; Qx=Aȳ) task step by step:
- 1
Task
A T-section base is 60×20 mm with y1=10 mm; its centered web is 20×40 mm with y2=40 mm. Find A, Qx about the bottom edge and ȳ.
Solution path
A1=1200 mm2, A2=800 mm2, A=2000 mm2.
Qx=1200·10+800·40=44,000 mm3 and ȳ=Qx/A=22 mm.
Check: A ȳ=44,000 mm3.
Frequently asked questions about Area centroid and first moment of area
How do I find a composite-section centroid?+
Split the area into simple, disjoint shapes and record each area Ai and centroid coordinates xi, yi. Then compute x̄=ΣAixi/ΣAi and ȳ=ΣAiyi/ΣAi. Treat holes as negative areas. In the T-section example, 1200 mm2 at 10 mm and 800 mm2 at 40 mm give a first moment of 44,000 mm3; divide by 2000 mm2 to obtain ȳ=22 mm. Use one coordinate system and nonoverlapping component areas; holes count negatively. Here Qx=∫y dA and Qy=∫x dA. First moment has m3, while I has m4.
What is the first moment of area, and what is it not?+
Here the first moment about the x-axis is Qx=∫y dA; it is the area-weighted position relative to the axis and has units m3. It vanishes about the centroidal axis. The second moment I=∫y2dA has units m4 and weights area by squared distance; it describes bending stiffness. Because terms such as “static moment” can be used inconsistently, state the integral, axis and unit explicitly. Use one coordinate system and nonoverlapping component areas; holes count negatively. Here Qx=∫y dA and Qy=∫x dA. First moment has m3, while I has m4.
How is this different from the parallel-axis theorem?+
First locate the centroid using first moments or area-weighted component centroids. Only then does the parallel-axis theorem transfer a second moment between parallel axes: I=Ic+Ad2. It does not calculate x̄ or ȳ. For a composite beam, both methods are often used in sequence: locate the centroid, then find I about the desired axis. Use one coordinate system and nonoverlapping component areas; holes count negatively. Here Qx=∫y dA and Qy=∫x dA. First moment has m3, while I has m4. Check the calculation with this case: A nonoverlapping T section has a 60×20 mm base, A1=1200 mm2 and y1=10 mm, plus a centered 20×40 mm web above it, A2=800 mm2 and y2=40 mm. Symmetry gives x̄=30 mm. About the bottom edge, Qx=1200·10+800·40=44000 mm3 and A=2000 mm2, so ȳ=22 mm. Check: Aȳ=2000·22=44000 mm3; the centroid lies within the 20–60 mm section height. Keep the model limits and units in view.
How do I check the result and units for Area centroid and first moment of area?+
List quantities, units and reference axis first: A in m2; x̄, ȳ, xi, yi in m; Qx and Qy in m3. Convert consistently and keep the axis convention explicit. Check the arithmetic with this example: A nonoverlapping T section has a 60×20 mm base, A1=1200 mm2 and y1=10 mm, plus a centered 20×40 mm web above it, A2=800 mm2 and y2=40 mm. Symmetry gives x̄=30 mm. About the bottom edge, Qx=1200·10+800·40=44000 mm3 and A=2000 mm2, so ȳ=22 mm. Check: Aȳ=2000·22=44000 mm3; the centroid lies within the 20–60 mm section height. Confirm that the units and centroid position are plausible; round only at the end. Use one coordinate system and nonoverlapping component areas; holes count negatively. Here Qx=∫y dA and Qy=∫x dA. First moment has m3, while I has m4.
Which model limit matters for Area centroid and first moment of area?+
The background states this starting condition: For a composite planar section, add nonoverlapping component areas with signs; holes are negative areas. The centroid follows. That limits when the result is a valid model prediction. Compare the problem’s boundary conditions with this assumption before substituting values; real components, material properties or loads can introduce additional effects. The cited reference supports the relation used here but does not remove the need to check whether the specific case fits the model. Use one coordinate system and nonoverlapping component areas; holes count negatively. Here Qx=∫y dA and Qy=∫x dA. First moment has m3, while I has m4.
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