Statics

The Centroid: A Shape's Balance Point

Centroid = balance point, the area average, on axes of symmetry, x̄ = ΣAx/ΣA weighted average, composite split, standard-shape centroids

Every area has a single balance point - its centroid, the average position of all its bits of area. Find it from symmetry when you can, and otherwise from the area-weighted average x̄ = Σ(A x)Σ A.

Drag the pivot. A flat shape balances only when the support sits exactly under its centroid; anywhere else it tips. That balance point is what the centroid means.

Switch shapes. Any axis of symmetry must pass through the centroid, so a symmetric shape hands you its center for free - no integral needed.

Drag the right area. The centroid is the area-weighted average x̄ = Σ(A x)ΣA, so it always leans toward the larger piece.

Split a complex shape into simple rectangles, locate each one's centroid, then combine them with the same weighted average. Drag a dimension and watch the whole centroid shift.

Memorize a few. A rectangle balances at its center, a triangle one-third up from its base, a semicircle a little above its flat edge. Toggle to see each.

In PracticeThe centroid is a shape's balance point, the average position of its area. Symmetry pins it down instantly - it always lies on every axis of symmetry - and otherwise you compute the area-weighted average x̄ = Σ(A x)ΣA, ȳ = Σ(A y)ΣA, which leans toward the larger pieces. Real shapes are handled by splitting them into simple parts (rectangle at its center, triangle a third up, semicircle just above its edge) and combining. This is exactly where a distributed load's resultant acts, and it sets up the next step - the area moment of inertia, which weights the same area by distance squared.

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