Area moment of inertia of a square lamina: For a square of side b, what is the second moment of area (area moment of inertia) about an axis through its centroid and parallel to one side?

Introduction / Context:
The area moment of inertia (second moment of area) quantifies a cross-section’s resistance to bending about a specified axis. For standard shapes like a square, knowing the centroidal moments is essential in beam design and structural analysis.

Given Data / Assumptions:

• Plane area is a square lamina of side b.
• Axis passes through the centroid (geometric centre).
• Axis is parallel to one side of the square (i.e., centroidal x- or y-axis).
• We are dealing with area (not mass) moment of inertia.

Concept / Approach:
Standard formula for a rectangle of breadth b and depth h about its centroidal axis parallel to breadth is I = bh^3/12. For a square b = h, so I = bb^3/12 = b^4/12. This holds for either principal centroidal axis due to symmetry (I_x = I_y).

Step-by-Step Solution:

Verification / Alternative check:
Polar moment at centroid J_o = I_x + I_y = b^4/12 + b^4/12 = b^4/6, consistent with symmetry and standard tables.

Why Other Options Are Wrong:

• b^3/4: wrong dimension (length^3 instead of length^4).
• b^4/3 or b^4/8: too large; would overestimate stiffness.
• b^4/36: centroidal about a diagonal is not b^4/36; this value does not correspond to a common centroidal axis for a square about a side-parallel axis.

Common Pitfalls:
Confusing area moment (length^4) with mass moment of inertia; using parallel-axis formula unnecessarily when the axis is through the centroid.

Final Answer:
b^4 / 12