centroids

A centroid is the geometric center of a shape or solid. For a plane region with uniform density, the centroid is the point where the region would balance if it were made of a thin, flat sheet. In three dimensions, the centroid serves as the center of mass for a solid body with uniform density and is the point where it would balance if supported at a single point.

In two dimensions, several cases are common. For a triangle with vertices at (x1, y1), (x2, y2), and (x3, y3), the centroid is the average of the vertex coordinates: ((x1 + x2 + x3)/3, (y1 + y2 + y3)/3). For a simple polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) taken in order, the polygon’s centroid can be computed from

Cx = (1/(6A)) sum over i of (xi + xi+1)(xi yi+1 − xi+1 yi),

Cy = (1/(6A)) sum over i of (yi + yi+1)(xi yi+1 − xi+1 yi),

where A is the polygon’s area and indices wrap around. More generally, the centroid of any plane region R with uniform density is given by the integral coordinates (x̄, ȳ) = (1/Area(R)) ∫∫R (x, y) dA. For a region defined by density ρ, the center of mass is (1/∫∫ρ dA) ∫∫ ρ (x, y) dA.

In three dimensions, the centroid extends to volume: (x̄, ȳ, z̄) = (1/Volume) ∭ (x, y, z) dV. Composite shapes have centroids equal to the weighted average of their parts’ centroids, weighted by area or volume.

Centroids are invariant under translation and rotation and lie at symmetry centers for highly regular figures. They are widely used in engineering, computer graphics, and physics to analyze balance, stability, and first moments.