centre of mass of polygon

Let A1A2An be an n-gon (http://planetmath.org/PolygonMathworldPlanetmathPlanetmath) 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

OM=1ni=1nOAi. (1)

We can of course take especially  O=A1,  and thus


In the special case of the triangle ABC we have

AM=13(AB+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 tetrahedronMathworldPlanetmathPlanetmath ABCD,


and any n-dimensional simplex (cf. the midpointMathworldPlanetmathPlanetmathPlanetmath (http://planetmath.org/Midpoint) of line segmentMathworldPlanetmath:  AM=12AB).

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