# centre of mass of polygon

Let $A_{1}A_{2}{\ldots}A_{n}$ be an $n$-gon (http://planetmath.org/Polygon) which is supposed to have a surface-density in all of its points, $M$ the centre of mass of the polygon and $O$ the origin. Then the position vector of $M$ with respect to $O$ is

 $\displaystyle\overrightarrow{OM}=\frac{1}{n}\sum_{i=1}^{n}\overrightarrow{OA_{% i}}.$ (1)

We can of course take especially  $O=A_{1}$,  and thus

 $\overrightarrow{A_{1}M}=\frac{1}{n}\sum_{i=1}^{n}\overrightarrow{A_{1}A_{i}}=% \frac{1}{n}\sum_{i=2}^{n}\overrightarrow{A_{1}A_{i}}.$

In the special case of the triangle $ABC$ we have

 $\displaystyle\overrightarrow{AM}=\frac{1}{3}(\overrightarrow{AB}+% \overrightarrow{AC}).$ (2)

The centre of mass of a triangle is the common point of its medians.

Remark. An analogical result with (2) concerns also the tetrahedron $ABCD$,

 $\overrightarrow{AM}=\frac{1}{4}(\overrightarrow{AB}+\overrightarrow{AC}+% \overrightarrow{AD}),$

and any $n$-dimensional simplex (cf. the midpoint (http://planetmath.org/Midpoint) of line segment:  $\overrightarrow{AM}=\frac{1}{2}\overrightarrow{AB}$).

 Title centre of mass of polygon Canonical name CentreOfMassOfPolygon Date of creation 2013-03-22 17:33:13 Last modified on 2013-03-22 17:33:13 Owner pahio (2872) Last modified by pahio (2872) Numerical id 11 Author pahio (2872) Entry type Result Classification msc 51P05 Classification msc 51M04 Classification msc 26B15 Classification msc 15A72 Synonym centroid of polygon Related topic ArithmeticMean Related topic AreaOfPolygon Related topic CentreOfMassOfHalfDisc Related topic BarycentricSubdivision Related topic CoordinatesOfMidpoint