Centroids and Centers of Gravity
Centroids of Composite Figures
$W \, \bar{y} = \Sigma wy$
$A \, \bar{y} = \Sigma ay$
$L \, \bar{y} = \Sigma ly$
Center of Gravity of Bodies and Centroids of Volumes
$W \, \bar{y} = \Sigma wy$
$W \, \bar{z} = \Sigma wz$
$V \, \bar{y} = \Sigma vy$
$V \, \bar{z} = \Sigma vz$
Centroids Determined by Integration
$\displaystyle A \, \bar{y} = \int_a^b y_c \, dA$
$\displaystyle L \, \bar{y} = \int_a^b y_c \, dL$
$\displaystyle W \, \bar{y} = \int_a^b y_c \, dW$
$\displaystyle W \, \bar{z} = \int_a^b z_c \, dW$
$\displaystyle V \, \bar{y} = \int_a^b y_c \, dV$
$\displaystyle V \, \bar{z} = \int_a^b z_c \, dV$
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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