Centroid of a region bounded by a polygon

Geometry Level pending

A region in the x y xy plane is bounded by a polygon with vertices { A i , i = 1 , 2 , . . . , N } \{\mathbf{A}_i, i = 1, 2, ..., N \} . Is it true that the centroid G \mathbf{G} of the region is given by the formula below ?

G = 1 N ∑ i = 1 N A i \mathbf{G} = \displaystyle \dfrac{1}{N} \sum_{i=1}^{N} \mathbf{A}_i

True False

Hosam Hajjir by Hosam Hajjir , 56, Canada

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1 solution

Mark Hennings
Dec 23, 2020

The formula is true for a triangular region. In particular the triangle O A B OAB has centroid g = 1 3 ( 0 + a + b ) = 1 3 ( a + b ) \mathbf{g} \; =\; \tfrac13(\mathbf{0} + \mathbf{a} + \mathbf{b}) \; = \; \tfrac13(\mathbf{a} + \mathbf{b}) But the triangle O A B OAB is also bounded by the polygon O A M B OAMB , where M M is the midpoint of A B AB , and 1 4 ( 0 + a + m + b ) = 3 8 ( a + b ) ≠ g \tfrac14(\mathbf{0} + \mathbf{a} + \mathbf{m} + \mathbf{b}) \; = \; \tfrac38(\mathbf{a} + \mathbf{b}) \; \neq \mathbf{g} unless O , A , B O,A,B are collinear. The desired formula does not hold in general.

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