Editing Centroid (section)

=== Of a polygon ===
The centroid of a non-self-intersecting closed [[polygon]] defined by <math>n</math> vertices <math>(x_0, y_0),\;</math><math>(x_1, y_1),\; \ldots,\;</math><math>(x_{n-1}, y_{n-1}),</math> is the point <math>(C_x, C_y),</math><ref name=bourke /> where

<math display=block>
C_{\mathrm x} = \frac{1}{6A}\sum_{i=0}^{n-1}(x_i+x_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i),</math>

and

<math display=block>
C_{\mathrm y} = \frac{1}{6A}\sum_{i=0}^{n-1}(y_i+y_{i+1})(x_i\ y_{i+1} - x_{i+1}\ y_i),
</math>

and where <math>A</math> is the polygon's signed area,<ref name=bourke>{{harvtxt|Bourke|1997}}</ref> as described by the [[shoelace formula]]:

<math display=block>
A = \frac{1}{2}\sum_{i=0}^{n-1} (x_i\ y_{i+1} - x_{i+1}\ y_i).
</math>

In these formulae, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex <math>(x_n, y_n)</math> is assumed to be the same as <math>(x_0, y_0),</math> meaning <math>i+1</math> on the last case must loop around to <math>i=0.</math> (If the points are numbered in clockwise order, the area <math>A,</math> computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)

{{anchor|Of a vertex set}}The centroid of a non-triangular polygon is not the same as its ''vertex centroid'', considering only its [[Vertex (geometry)|vertex]] set (as the [[#Of a finite set of points|centroid of a finite set of points]]; {{xref|see also: [[Polygon#Centroid]]}}).