# 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

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?