In one line
Each primitive shape has a fixed, memorised centroid location measured from a known edge or corner, which you use as building blocks for composite figures.
Explained simply
Just like you memorise multiplication tables before doing long sums, you memorise the centroid of a few basic shapes. Every MU centroid problem is just these blocks added or cut out.
Technically, these values come from integration and are measured from the reference edges shown below.
How it works
- Identify the primitive shape and its orientation in the figure.
- Pick the value from the table, measured from the stated edge or corner.
- Convert it into coordinates of your chosen x-y axes for the full figure.
| Shape | Area | Centroid |
|---|---|---|
| Rectangle b × h | \( bh \) | \( b/2 \) and \( h/2 \) from the sides |
| Triangle, base b, height h | \( bh/2 \) | \( h/3 \) from the base (2h/3 from the apex) |
| Right triangle | \( bh/2 \) | \( b/3 \), \( h/3 \) from the right-angle corner |
| Circle radius r | \( \pi r^2 \) | at the centre |
| Semicircle radius r | \( \pi r^2/2 \) | on the axis of symmetry, \( 4r/3\pi \) from the diameter |
| Quarter circle radius r | \( \pi r^2/4 \) | \( 4r/3\pi \) from each straight edge |
Key formula
\( \dfrac{4r}{3\pi} \approx 0.424r \) for semicircles and quarter circles; triangles: \( h/3 \) from the base. All lengths in mm, areas in mm\(^2\).
Quick example
A semicircle of radius 30 mm sits on top of a rectangle, with its diameter on the rectangle's top edge at y = 50 mm. Its centroid is at \( y = 50 + \dfrac{4(30)}{3\pi} = 50 + 12.73 = 62.73 \) mm, and its area is \( \pi(30)^2/2 = 1413.7 \) mm\(^2\).
Common mistakes
- Measuring \( h/3 \) from the apex instead of the base of a triangle.
- Measuring \( 4r/3\pi \) from the curved edge instead of the straight diameter.
- Forgetting to add the offset of the shape's edge from your reference axis.
How MU asks it
These values are used in every Module 2 numerical. Sometimes a 3 to 4-mark part asks you to list them or derive one (see the derivation questions). In the numerical, write a table with columns A, x, y, Ax, Ay so the examiner can see exactly which standard value you used.