Centroid and "Centre of mass"

Question

I thought that the centroid was also the point at which the area in all directions was net 0. Aka area above the cenrroid was the same area as below.

But for a triangle everything I can see seems to suggest that the area above the centroid of a triangle is about 55% of the area, I just dont see how this fits the fact of it being the "centre of mass"

Even integrating it or just simple equations (rearranging the area of a traingle formula) and even when I checked in Geogebra seemed to show that the point at which half the area is reached vertically is ${\sqrt2}/2$ from the apex for a unit triangle.

I am so sure I'm wrong but I cant see how I could be

Answer 1

John Conway and others discussed this is 2001 and I produced a brief web page illustrating this with a no-longer functional Java applet. You are correct that apart from the three medians (blue below), none of the area bisectors (green below) pass through the centroid.

But if the triangle is solid then any line through the centroid will balance the triangle, in the sense that the first moments on each side will be equal and opposite, even if the two areas are different; regions of the same area but further away will have a greater moment.

Answer 2

The centroid is the point at which the figure will balance when you prop it up on a pin.

As you note, that does not mean that every line through the centroid bisects the area. Balancing takes into account the distances from the balance point to the points of the figure.

If you think about this a bit you will see that very few figures have a point such that every line through that point bisects the area. (Probably only circles do - I don't know a proof offhand.)

https://en.wikipedia.org/wiki/Centroid