Area moments – rectangle about centroidal axis versus base For a rectangle (width b, height h), what is the ratio of its second moment of area about the centroidal horizontal axis to that about its base?

Introduction / Context:
Second moment of area (area moment of inertia) determines bending stiffness and deflection of beams. For rectangles, knowing values about different axes is crucial for quick checks and design sizing in structural engineering.

Given Data / Assumptions:

• Rectangle width b and height h.
• Centroidal horizontal axis (through the centroid parallel to base).
• Base axis coincident with the rectangle's base edge.

Concept / Approach:

Standard results: I_centroid = (b * h^3) / 12 about the centroidal horizontal axis. Using the parallel-axis theorem to the base gives I_base = I_centroid + A * d^2, where A = b * h and d = h/2. Thus I_base = (b * h^3)/12 + (b * h) * (h/2)^2 = (b * h^3)/3. The requested ratio is (1/12) / (1/3) = 1/4.

Step-by-Step Solution:

Verification / Alternative check:

Dimensional consistency: both moments scale with b * h^3, so the ratio is dimensionless and independent of size.

Why Other Options Are Wrong:

(b), (c), (d), and (e) do not match the known values derived from standard formulas and the parallel-axis theorem.

Common Pitfalls:

Mixing up centroidal and base axes; forgetting the parallel-axis term A * d^2; confusing I_x and I_y orientations.

Final Answer:

1/4