The center of gravity of a hexagon

Question

Good morning everyone,

I am doing some researches and I wanted to have a proper answer concerning the centroid of a hexagon. Is it equal to the arithmetic mean of its vertex coordinates? And is it the the same for any other sort of polygon ?

Thank you in advance.

Answer 1

The centroid of a hexagon considered as a filled "object" (think to a cardboard hexagon) has a surface centroid $C_s$ which is different in general from the centroid $C_v$ of its vertices, and different also from the centroid $C_e$ of its edges (think to metal bars materializing the edges). You have 3 different kind of centroids (2 of them can coincide in very special cases such as a hexagon with a symmetry line).

You ask for other polygons : we have the same behavior in general for any polygon (3 different centroids) excepted the triangle, where the surface centroid $C_s$ is the same as the vertices centroid $C_v$ (aka center of gravity) but not the same as the edges centroid $C_e$ in general.