## Centroid Area

For the below figure, what is the y-coordinate of the centroid?

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__**Hint**

**Hint**

Split the object into basic shapes (triangles, rectangles, squares, etc.).

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__**Hint 2**

**Hint 2**

The y-component of an area’s centroid is defined as:

$$$y_{ac}=\frac{\sum y_n a_n}{A}$$$

where
$$a_n$$
is the simplified object’s area,
$$y_n$$
is the distance from the origin to the middle of the simplified object, and
$$A$$
is the total area.

The y-component of an area’s centroid is defined as:

$$$y_{ac}=\frac{\sum y_n a_n}{A}$$$

where
$$a_n$$
is the simplified object’s area,
$$y_n$$
is the distance from the origin to the middle of the simplified object, and
$$A$$
is the total area (
$$\Sigma a_n$$
). First, let’s split the main object into basic shapes (triangles, rectangles, squares, etc.).

To find the y-component of our main object’s centroid, let’s set the ground as the origin:

$$$y_{ac}=\frac{(y_A a_A)+(y_B a_B)}{(a_A+a_B)}=\frac{[(\frac{3m}{2}+6m)(4m\cdot 3m)]+[\frac{6m}{2}\cdot(6m\cdot 1m)]}{[(4m\cdot 3m)+(6m\cdot 1m)]}$$$

$$$=\frac{[(7.5m)(12m^2)+(3m)(6m^2)]}{(12m^2+6m^2)}=\frac{(90m^3+18m^3)}{18m^2}=\frac{108m^3}{18m^2}=6\:m$$$

6 m