Parallel-Axis Theorem for Area Moments of Inertia According to the parallel-axis theorem, the moment of inertia IP of a plane area about any axis parallel to an axis through its centroid (C.G.) is:

Introduction / Context:
The parallel-axis theorem is a cornerstone of mechanics of materials and structural analysis. It allows engineers to compute the second moment of area (area moment of inertia) about any axis parallel to a centroidal axis without direct integration.

Given Data / Assumptions:

• IG: centroidal area moment of inertia.
• A: area of the section.
• h: perpendicular distance between the centroidal axis and the new parallel axis.

Concept / Approach:
The theorem states that when shifting from the centroidal axis to a parallel axis a distance h away, the new moment equals the centroidal moment plus A times the square of that distance. This arises from the binomial expansion of (y + h)^2 during area integration.

Step-by-Step Solution:
Start from IP = ∫(y + h)^2 dA. Expand: ∫(y^2 + 2yh + h^2) dA = ∫y^2 dA + 2h ∫y dA + h^2 ∫dA. Because the centroid is on the axis, ∫y dA = 0. Thus IP = IG + h^2 A.

Verification / Alternative check:
Dimensional check: IG has units of length^4; A * h^2 also has length^4, so the sum is dimensionally consistent.

Why Other Options Are Wrong:
Subtracting A h^2 contradicts the derivation (moment increases as axis moves away). Division or inverse forms lack physical and dimensional meaning.

Common Pitfalls:
Confusing the area theorem with the mass moment version (form is analogous but quantities differ). Using wrong distance (must be perpendicular between parallel axes).

Final Answer:
IP = IG + A * h^2