Finding the combined centroid of two regions given the centroids of both regions

1. Feb 7, 2013

theBEAST

1. The problem statement, all variables and given/known data

3. The attempt at a solution
I tried to do this problem by taking the average of the y-bar centroid values but that gave me the wrong answer. I am only interesting in knowing why this method is incorrect.

Thanks!

2. Feb 7, 2013

SteamKing

Staff Emeritus
What is the definition of centroid? Your method would be valid if the areas of the two regions were equal.

3. Feb 7, 2013

HallsofIvy

Staff Emeritus
It can be shown that the centroid of the union of two (non-overlapping) regions is the weighted average of the two centroids, weighted by the areas of the regions. That is, if the two regions have centroid $(x_1, y_1)$ and $(x_2, y_2)$ and have areas $A_1$ and $A_2$, respectively, then the centroid of the combined regions is at
$$\left(\frac{A_1x_1+ A_2x_2}{A_1+ A_2}, \frac{A_1y_1+ A_2y_2}{A_1+ A_2}\right)$$

4. Feb 7, 2013

haruspex

Re: Finding the combined centroid of two regions given the centroids o

As your grandmother should have taught you, don't take an average of averages. If an elephant and a flea get on opposite ends of a symmetric seesaw, will they balance? So where's their combined centroid?