# Center of Mass of Quadrilateral

I recently started studying Mass Points and the question arose:

If you have a quadrilateral with a mass of 1 at each vertex, how do you locate the center of mass.

I had several approaches but I was not sure if they all result in the same point or not. My approaches are:

1. With quadrilateral ABCD, find the center of mass of triangle ABC, which will have weight 3, and the center of mass of the quadrilateral will be 1/4 of the way into the segment from the centroid of ABC to point D.

2. Connect the centroids of ABC and BCD and the centroids of ABD and ACD. The intersection will be the center of mass of ABCD.

3. Connect the midpoints of AB and CD and the midpoints of BC and AD. The intersection will be the center of mass of ABCD.

Are these three points the same?

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How do you find it algebraically or geometrically? –  ja72 Dec 27 '13 at 2:31

A hint:

Call your points $P_i=(x_i,y_i)$ $\ (1\leq i\leq 4)$ and compute the center $C^{(j)}$ for each of your ${\rm approaches}_j$ $\ (1\leq j\leq3)$ coordinate-wise.

Note that the center of mass of the four vertices in general does not coincide with the center of mass of the quadrilateral surface determined by these vertices. For a triangle these two centers coincide.

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If you have any set of mass points with masses $m_k$ at vertices $P_k = (x_k,y_k)$ then the center of mass is given by: $$C_x = \frac{\sum_{k=1}^n x_k m_k}{\sum_{k=1}^n m_k} \qquad ; \qquad C_y = \frac{\sum_{k=1}^n y_k m_k}{\sum_{k=1}^n m_k}$$ Substitute $m_k = 1$ and $n = 4\,$ in this and you're just done. It becomes all more complicated if the mass points are not pinpointed at the vertices (e.g. distributed along edges or within an area).