Area moments – triangular section about its base\nThe second moment of area (area moment of inertia) of a triangular section with base b and height h about its base is:

Introduction / Context:
Area moments of inertia are central to bending stress and deflection analyses. For common shapes, standard formulas are used to speed design and checks.

Given Data / Assumptions:

• Plane area is a triangle of base b and height h.
• Axis of interest is along the base line of the triangle (in-plane axis through the base).
• Small-deflection, linear elasticity context (typical beam theory usage).

Concept / Approach:
The area moment of inertia about the base for a triangle is derived by integrating y^2 dA from the base toward the apex, using a linear variation of width with height.

Step-by-Step Solution:

Verification / Alternative check:
Parallel-axis transformation from centroidal axis (I_cg = b h^3 / 36) to base: I_base = I_cg + A d^2 with d = h/3 and A = b h / 2. This also yields b h^3 / 12.

Why Other Options Are Wrong:

• b h^3 / 4 and b h^3 / 8 are too large; they correspond to different shapes/axes.
• b h^3 / 36 is the centroidal moment for a triangle about a base-parallel centroidal axis, not about the base itself.

Common Pitfalls:
Mixing centroidal and base axes or forgetting linear width variation.

Final Answer:
b h^3 / 12