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Centroid of a Composite Area

x ˉ = \frac{\sum A _{i} x _{i}}{\sum A _{i}}, y ˉ = \frac{\sum A _{i} y _{i}}{\sum A _{i}}

The centroid is the geometric center. For composites, use the p. 99–100 area table (rectangle at b/2, h/2; right-triangle height centroid h/3 from the base; semicircle 4a/(3π) from the diameter) and treat holes as negative areas.

A_{i}

(m²)

Area of part i

x_{i}, y_{i}

(m)

Centroid coordinates of part i

Step-by-step solved example

An L-section is a 200×20 mm vertical stem plus a 120×20 mm base to the right, flush at the bottom. Find ȳ from the bottom.

Composite-area centroid: Σ(Ai ȳi)/ΣAi. Split T or L shapes into rectangles.

1. Areas
A1 (stem) = 200×20 = 4000 mm², y1 = 100 mm. A2 (base exclusive of stem overlap) = 100×20 = 2000 mm², y2 = 10 mm. (Use non-overlapping parts.)
2. Centroid
ȳ = (4000·100 + 2000·10)/6000 = 420000/6000 = 70 mm.
$y ˉ = 70 mm$

Answer: ȳ = 70 mm from the bottom

10 practice questions

0/10 correct

1.Centroid of a rectangle is at

2.Triangle centroid is located

3.A 4×6 rectangle, origin at a corner: ȳ =

4.A hole is treated as

5.Semicircle of radius R: centroid from diameter is

6.Two equal squares side-by-side: composite centroid is

7.First moment of area Q about x is

8.Parallel-axis theorem for I uses

9.A 3 m² area with centroid 0.4 m from an axis: Q =

10.T-beam flange 200×20 and web 20×180 (below flange). ȳ from top is nearest

Given T-section: flange 200 by 20 on top, web 20 by 180 below the flange. y is measured from the top.