Geometric identities for spheres: which expression corresponds to the surface-based shape factor for a sphere when using particle diameter D (with A = surface area, V = volume)?

Introduction / Context:
Particle technology uses simple geometric relations for ideal shapes to define size descriptors, sphericity, and shape factors. For spheres, exact formulas exist linking surface area (A), volume (V), and diameter (D). Recognizing these identities is fundamental before extending to irregular particles and shape-corrected diameters (e.g., volume-surface mean diameter).

Given Data / Assumptions:

• Spherical particle of diameter D.
• Surface area A and volume V denote geometric properties.
• We are asked specifically about a surface-based factor.

Concept / Approach:
For a sphere, A = π D^2 and V = (π/6) D^3. The commonly referenced “surface shape factor” in simple MCQs is A / D^2, which for a sphere equals π. The companion “volume shape factor” is V / D^3 = π/6. Both are exact identities that act as baselines for comparing non-spherical particles.

Step-by-Step Solution:
Recall area of sphere: A = 4 * π * r^2 = π * D^2.Compute A / D^2 = (π * D^2) / D^2 = π.Recognize that V = π/6 * D^3 is the volume identity, not the requested surface-based factor.

Verification / Alternative check:
Dimensional analysis confirms A / D^2 is dimensionless; for a sphere, this reduces exactly to the constant π, matching textbook geometry.

Why Other Options Are Wrong:
π/6 (= V / D^3) is the volume-based identity, not the surface shape factor asked for.“None of these” is incorrect because π (= A / D^2) is correct.

Common Pitfalls:
Mixing up surface and volume relations; ensure you identify whether the factor normalizes area or volume before selecting the expression.

Final Answer:
π (= A / D^2)