Second moment of area of a triangular section For a triangle of base b and height h, find the second moment of area about the centroidal axis parallel to the base.

Introduction / Context:
The centroidal second moment of area of a triangle about an axis parallel to its base is a standard result used in beam bending calculations and deflection analysis.

Given Data / Assumptions:

• Triangle of base b and height h.
• Axis: through centroid, parallel to base.
• Homogeneous thin area (geometric property).

Concept / Approach:
Starting from the base axis, the area moment about the base is I_base = b h^3 / 12. The centroidal axis is located at a distance h/3 from the base. Use the parallel-axis theorem to move to the centroidal axis.

Step-by-Step Solution:

Verification / Alternative check:
Dimension check yields length^4. The value is smaller than I about the base, as expected.

Why Other Options Are Wrong:

• b h^3/12: About the base, not centroid.
• b h^3/24, /48, /6: Incorrect constants from misapplication of the parallel-axis theorem.

Common Pitfalls:
Using base-axis value directly for centroid; forgetting that A = (1/2) b h for a triangle.

Final Answer:
I = b h^3 / 36