## Centroids of Composite Figures

Center of gravity of a homogeneous flat plate
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

Centroids of areas
$A \, \bar{x} = \Sigma ax$

$A \, \bar{y} = \Sigma ay$

Centroids of lines
$L \, \bar{x} = \Sigma lx$

$L \, \bar{y} = \Sigma ly$

## Center of Gravity of Bodies and Centroids of Volumes

Center of gravity of bodies
$W \, \bar{x} = \Sigma wx$

$W \, \bar{y} = \Sigma wy$

$W \, \bar{z} = \Sigma wz$

Centroids of volumes
$V \, \bar{x} = \Sigma vx$

$V \, \bar{y} = \Sigma vy$

$V \, \bar{z} = \Sigma vz$

## Centroids Determined by Integration

Centroid of area
$\displaystyle A \, \bar{x} = \int_a^b x_c \, dA$

$\displaystyle A \, \bar{y} = \int_a^b y_c \, dA$

Centroid of lines
$\displaystyle L \, \bar{x} = \int_a^b x_c \, dL$

$\displaystyle L \, \bar{y} = \int_a^b y_c \, dL$

Center of gravity of bodies
$\displaystyle W \, \bar{x} = \int_a^b x_c \, dW$

$\displaystyle W \, \bar{y} = \int_a^b y_c \, dW$

$\displaystyle W \, \bar{z} = \int_a^b z_c \, dW$

Centroids of volumes
$\displaystyle V \, \bar{x} = \int_a^b x_c \, dV$

$\displaystyle V \, \bar{y} = \int_a^b y_c \, dV$

$\displaystyle V \, \bar{z} = \int_a^b z_c \, dV$

Centroids of Common Geometric Shapes

 Rectangle Area and Centroid $A = bd$ $\bar{x} = \frac{1}{2}b$ $\bar{y} = \frac{1}{2}d$

 Triangle Area and Centroid $A = \frac{1}{2}bh$ $\bar{y} = \frac{1}{3}h$

 Circle Area and Centroid $A = \pi r^2$ $\bar{x} = 0$ $\bar{y} = 0$

 Semicircle Area and Centroid $A = \frac{1}{2}\pi r^2$ $\bar{x} = 0$ $\bar{y} = \dfrac{4r}{3\pi}$

 Semicircular Arc Length and Centroid $L = \pi r$ $\bar{x} = \dfrac{2r}{\pi}$ $\bar{y} = 0$

 Quarter Circle Area and Centroid $A = \frac{1}{4}\pi r^2$ $\bar{x} = \dfrac{4r}{3\pi}$ $\bar{y} = \dfrac{4r}{3\pi}$

 Sector of a Circle Area and Centroid $A = r^2 \theta_{rad}$ $\bar{x} = \dfrac{2r \sin \theta}{3\theta_{rad}}$ $\bar{y} = 0$

 Circular Arc Length and Centroid $L = 2r \theta_{rad}$ $\bar{x} = \dfrac{r \sin \theta}{\theta_{rad}}$ $\bar{y} = 0$

 Ellipse Area and Centroid $A = \pi ab$ $\bar{x} = 0$ $\bar{y} = 0$

 Half Ellipse Area and Centroid $A = \frac{1}{2}\pi ab$ $\bar{x} = 0$ $\bar{y} = \dfrac{4b}{3\pi}$

 Quarter Ellipse Area and Centroid $A = \frac{1}{4}\pi ab$ $\bar{x} = \dfrac{4a}{3\pi}$ $\bar{y} = \dfrac{4b}{3\pi}$

 Parabolic Segment Area and Centroid $A = \frac{2}{3} bh$ $\bar{x} = \frac{3}{8}b$ $\bar{y} = \frac{2}{5}h$

 Spandrel Area and Centroid $A = \dfrac{1}{n + 1} bh$ $\bar{x} = \dfrac{1}{n + 2}b$ $\bar{y} = \dfrac{n + 1}{4n + 2}h$

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