I was having a look at this question on SO.

From what I know, the centroid is the center of mass of an object. so, by definition its position is given by a simple mean of the positions of all the points in the object.

For a polygon, it only has mass at the vertices. So, the centroid should be given by the arithmetic mean of the coordinates of the vertices.

But Wikipedia says centroid is given by



where A is



Why doesn't a simple arithmetic mean work?

The centroid of a polygon is indeed its center of mass -- but the mass of a polygon is uniformly distributed over its surface, not only at the vertices. You're right that if the mass were split evenly among the vertices only, the centroid would be the arithmetic mean of the coordinates of the vertices.

It just so happens that both definitions are equivalent (mass evenly distributed over the surface vs mass at the vertices only) for simple shapes like triangles and rectangles.
