In one line
Integrate thin horizontal strips parallel to the base; the first moment divided by the area gives the centroid at one-third of the height from the base.
Explained simply
Slice a triangle into very thin horizontal strips like slices of a sandwich. Strips near the base are long, strips near the tip are short. Because more area sits near the base, the balancing line is closer to the base, at \( h/3 \).
Technically, we use \( \bar{y} = \int y\,dA / \int dA \) with a strip of width x and thickness dy at height y above the base.
How it works
- Take a triangle of base b and height h, with the base on the x-axis.
- At height y, by similar triangles the strip length is \( x = \dfrac{b(h-y)}{h} \).
- Strip area \( dA = x\,dy = \dfrac{b}{h}(h-y)\,dy \).
- Total area \( A = \int_0^h \dfrac{b}{h}(h-y)\,dy = \dfrac{bh}{2} \).
- First moment \( \int y\,dA = \dfrac{b}{h}\int_0^h (hy - y^2)\,dy = \dfrac{b}{h}\left(\dfrac{h^3}{2} - \dfrac{h^3}{3}\right) = \dfrac{bh^2}{6} \).
- \( \bar{y} = \dfrac{bh^2/6}{bh/2} = \dfrac{h}{3} \) from the base.
- Since any side can be the base, the centroid lies at the intersection of the medians.
Key formula
\( \bar{y} = \dfrac{\int_0^h y\,dA}{\int_0^h dA} = \dfrac{h}{3} \), where h is the perpendicular height (mm) and b the base (mm). The result does not depend on b.
Quick example
For a triangular plate with base 90 mm and height 60 mm, the centroid is \( 60/3 = 20 \) mm above the base, i.e. 40 mm below the apex. Its area is \( 90 \times 60/2 = 2700 \) mm\(^2\).
Common mistakes
- Writing the strip length as \( by/h \) while measuring y from the base; that length is correct only with y measured from the apex, and mixing them gives the wrong answer \( 2h/3 \).
- Forgetting to divide by the total area at the end.
- Using wrong integration limits.
How MU asks it
Asked as a 4 to 5-mark derivation: 'Determine the centroid of a triangle by the method of integration'. Draw the figure with the strip, write the similar-triangle relation, integrate neatly, and box the answer \( h/3 \) from the base.