Centroid of a Trapezium For a trapezium of height h with parallel sides a (top) and b (base), the centroid lies at a distance y from the base b. What is y?

Introduction / Context:
Locating the centroid (centre of gravity for a lamina of uniform density) is fundamental in mechanics and structural analysis. For trapezoids (trapezia), the centroid position along the height depends on the relative sizes of the parallel sides.

Given Data / Assumptions:

• Trapezium with parallel sides a (top) and b (base), separated by height h.
• Uniform thickness and density (so centroid = centre of area).
• Distance y is measured upward from the base b toward the top side a.

Concept / Approach:
The centroid of a trapezium can be obtained using the first-moment (area * distance) method or standard compiled results. When measured from the base b toward the smaller side a, the centroid lies closer to the larger base and is given by a weighted ratio favouring the larger side length in the numerator.

Step-by-Step Solution:

Verification / Alternative check:
Special case a = b (rectangle): y = h * (2a + a) / [3 * (2a)] = h * 3a / (6a) = h / 2, which agrees with a rectangle’s centroid. If a → 0 (triangle), y → h * b / (3b) = h / 3 from the base, matching the triangle result.

Why Other Options Are Wrong:

• y = h * (a + 2b) / [3(a + b)] is the expression measured from the top when roles of a and b swap; not correct for distance from base b as posed.
• y = (h * (a + b)) / 2 is just mid-height, valid only for a = b.
• y = h * (b - a) / [3(a + b)] has incorrect dependence and can be negative.

Common Pitfalls:
Confusing which side is the reference (top vs base) and swapping a, b; forgetting that centroid shifts toward the larger parallel side.

Final Answer:
y = [h * (2a + b)] / [3 * (a + b)].