Section Properties Studio
Moment of inertia for the base shapes, and a composite-section builder: stack rectangles and circles (holes allowed), get the centroid and the composite Ixx by the parallel-axis theorem — drawn live with the centroid marked. Covers §10.0.
Base Shape: Ixx about the Centroid
Composite Section Builder
One shape per line: rect x y b h or circle x y d, where (x, y) is the shape's own centroid. Prefix with - to subtract a hole, e.g. -circle 0 30 12.
Composite section to scale; holes hatched; the red marker is the computed centroid. FIG. E09-1
The parallel-axis theorem is the whole trick: I = Σ(Īᵢ + Aᵢdᵢ²). Area far from the neutral axis contributes with the square of its distance — which is why an I-beam puts its material in the flanges and almost none in the web.