Editing Polygon (section)

===Centroid===
Using the same convention for vertex coordinates as in the previous section, the coordinates of the centroid of a solid simple polygon are 
:<math>C_x = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (x_i + x_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i), </math>
:<math>C_y = \frac{1}{6 A} \sum_{i = 0}^{n - 1} (y_i + y_{i + 1}) (x_i y_{i + 1} - x_{i + 1} y_i).</math>
In these formulas, the signed value of area <math>A</math> must be used.

For [[triangle]]s ({{math|1=''n'' = 3}}), the centroids of the vertices and of the solid shape are the same, but, in general, this is not true for {{math|''n'' > 3}}. The [[centroid]] of the vertex set of a polygon with {{mvar|n}} vertices has the coordinates
:<math>c_x=\frac 1n \sum_{i = 0}^{n - 1}x_i,</math>
:<math>c_y=\frac 1n \sum_{i = 0}^{n - 1}y_i.</math>