# Proof for Centroid Formula for a Polygon

I was reading a paper and I found this formula for the centroid of a polygon in terms of its coordinates but no proof was given.

$C_x =\frac{1}{6A} \sum_{i=0}^{N-1}(x_i+x_{i+1})(x_iy_{i+1}-x_{i+1}y_i)$

$C_y =\frac{1}{6A} \sum_{i=0}^{N-1}(y_i+y_{i+1})(x_iy_{i+1}-x_{i+1}y_i)$

where $x_N=x_0$ and A = the area of the polygon.

Does anyone know the proof for this formula? And how is it connected (if it is) to the formula for the centroid of a uniformly dense lamina:

Assume that the polygon is star-shaped with respect to the origin and that the vertices are consecutively numbered in a counterclockwise direction. The polygon can be decomposed into triangles defined by the origin and successive vertices $\mathbf v_i$ and $\mathbf v_{i+1}$. The area of each of these triangles is $\frac12(x_iy_{i+1}-x_{i+1}y_i)$. If we concentrate all of this “mass” at the centroid of the triangle $\mathbf c_i$, then the centroid of the polygon is given by the usual formula for the center of mass of a set of point masses: $$\mathbf c=\frac1A\sum_{i=0}^{N-1}\frac12(x_iy_{i+1}-x_{i+1}y_i)\mathbf c_i.\tag{1}$$ The centroid of a triangle with vertices $\mathbf u$, $\mathbf v$ and $\mathbf w$ is just $\frac13(\mathbf u+\mathbf v+\mathbf w)$. Substituting this into $(1)$ yields the formula in the question.
• Thanks so much! Just one more question, how does u+v+w translate into $x_i + x_{i+1}$ and similarly for y? – Gabriel Mar 18 '16 at 2:11
• @Gabriel $\mathbf u=(0,0)$, $\mathbf v = (x_i,y_i)$, $\mathbf w=(x_{i+1},y_{i+1})$. – amd Mar 18 '16 at 2:13