Area properties: Polar moment of inertia of a circular section about an axis perpendicular to the plane through its centroid. State the correct expression in terms of radius r or diameter d. Choose the correct option.

Given/Definitions

• For a circular area of radius r, the centroidal second moments are I_x = I_y = (π r⁴)/4.
• The polar moment about the perpendicular centroidal axis is J = I_x + I_y.

Step-by-step calculation
J = I_x + I_y = (π r⁴)/4 + (π r⁴)/4 = (π r⁴)/2. In terms of diameter d (r = d/2): J = (π (d/2)⁴)/2 = (π d⁴)/32.

Verification
Option (a) matches the r–form; the equivalent d–form is option (d).

Final Answer
J = (π r^4) / 2 (equivalently, (π d^4)/32).