For a cross-section of area A with least centroidal second moment of area I, what is the radius of gyration about that centroidal axis?

Introduction / Context:
The radius of gyration characterizes how area is distributed about an axis and is widely used in column buckling and section classification. It provides a compact measure linking area and second moment of area for stability checks.

Given Data / Assumptions:

• Cross-sectional area = A.
• Least centroidal second moment of area = I (about the governing axis).
• We seek the radius of gyration r about that centroidal axis.

Concept / Approach:
By definition, I = A * r^2, so r = sqrt(I / A). For buckling, the least radius of gyration controls Euler-type critical loads since it corresponds to the weakest bending axis.

Step-by-Step Solution:
Start from definition: I = A * r^2.Rearrange: r^2 = I / A.Take square root: r = sqrt(I / A).

Verification / Alternative check:
Dimensional check: [I] = L^4, [A] = L^2 → I/A = L^2 → sqrt yields L, consistent for a length measure.

Why Other Options Are Wrong:

• r = I / A and r = A / I: Wrong dimension (length vs. length squared or inverse).
• r = sqrt(A / I): Inverse of the correct relationship.
• r = I * A: Nondimensional nonsense for r.

Common Pitfalls:
Confusing section modulus (I / y) with radius of gyration; forgetting that r is a geometric length derived from I and A.

Final Answer:
r = sqrt(I / A)