Area Properties — Square Section (Centroidal Axis) What is the area moment of inertia of a square plate of side length a about an axis through its centre of gravity (centroid) and perpendicular to the plane of the square?

Introduction / Context:
Engineers frequently need the area moment of inertia (also called the second moment of area) of basic shapes to evaluate bending stress and deflection. For a square plate, the centroidal moments about in-plane axes are standard results used in beam and plate calculations.

Given Data / Assumptions:

• Shape: square, side length a.
• Axis: through the centroid, perpendicular to the plate for polar, but here we need the in-plane centroidal axis result commonly quoted as I_x or I_y for a square (either is the same by symmetry).
• Material distribution is uniform; this is an area property, not a mass property.

Concept / Approach:
The area moment of inertia quantifies how area is distributed relative to an axis. For a rectangle of breadth b and depth d, I_centroid about the axis parallel to b is (b d^3) / 12. Setting b = d = a for a square gives I = a^4 / 12 about either centroidal in-plane axis. The polar area moment about the perpendicular centroidal axis would be J = I_x + I_y = a^4 / 6, but that is not asked here.

Step-by-Step Solution:

Verification / Alternative check:
Compare with polar relation: J = I_x + I_y = 2 * (a^4 / 12) = a^4 / 6. This is a known closed-form identity for a square, confirming I_x = a^4 / 12.

Why Other Options Are Wrong:
a^4 / 4 and a^4 / 8 are too large and would overestimate stiffness; a^4 / 36 is the centroidal inertia of a triangle-like scaling, not applicable to a square; the correct rectangular/square formula yields a^4 / 12.

Common Pitfalls:
Confusing area moment (used in bending) with mass moment (used in dynamics); mixing the polar moment with the in-plane centroidal moment.

Final Answer:
a^4 / 12.