centre of mass of polygon

Let $A_1{A}_2\mathrm{\dots }{A}_n$ 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

$\overrightarrow{OM}={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n}\overrightarrow{O{A}_i}.$ (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

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