List of centroids

The following diagrams depict a list of centroids. A centroid of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X. For an object of uniform composition (mass, density, etc.) the centroid of a body is also its centre of mass.

Shape	Figure	$\bar x$	$\bar y$	Area
Right-triangular area		$\frac{-b}{3}$	$\frac{h}{3}$	$\frac{bh}{2}$
Quarter-circular area		$\frac{4r}{3\pi}$	$\frac{4r}{3\pi}$	$\frac{\pi r^2}{4}$
Semicircular area		$\,\!0$	$\frac{4r}{3\pi}$	$\frac{\pi r^2}{2}$
Quarter-elliptical area		$\frac{4a}{3\pi}$	$\frac{4b}{3\pi}$	$\frac{\pi a b}{4}$
Semielliptical area		$\,\!0$	$\frac{4b}{3\pi}$	$\frac{\pi a b}{2}$
Semiparabolic area	The area between the curve $y = \frac{h}{b^2} x^2$ and the $\,\!y$ axis, from $\,\!x = 0$ to $\,\!x = b$	$\frac{3b}{8}$	$\frac{3h}{5}$	$\frac{2bh}{3}$
Parabolic area	The area between the curve $\,\!y = \frac{h}{b^2} x^2$ and the line $\,\!y = h$	$\,\!0$	$\frac{3h}{5}$	$\frac{4bh}{3}$
Parabolic spandrel	The area between the curve $\,\!y = \frac{h}{b^2} x^2$ and the $\,\!x$ axis, from $\,\!x = 0$ to $\,\!x = b$	$\frac{3b}{4}$	$\frac{3h}{10}$	$\frac{bh}{3}$
General spandrel	The area between the curve $y = \frac{h}{b^n} x^n$ and the $\,\!x$ axis, from $\,\!x = 0$ to $\,\!x = b$	$\frac{n + 1}{n + 2} b$	$\frac{n + 1}{4n + 2} h$	$\frac{bh}{n + 1}$
Circular sector	The area between the curve (in polar coordinates) $\,\!r = \rho$ and the pole, from $\,\!\theta = -\alpha$ to $\,\!\theta = \alpha$	$\frac{2\rho\sin(\alpha)}{3\alpha}$	$\,\!0$	$\,\!\alpha \rho^2$
Circular segment		$\,\!0$	$\frac{4R\sin^3{\frac{\theta}{2}}}{3(\theta-\sin{\theta})}$	$\frac{R^2}{2}(\theta -sin{\theta})$
Quarter-circular arc	The points on the circle $\,\!x^2 + y^2 = r^2$ and in the first quadrant	$\frac{2r}{\pi}$	$\frac{2r}{\pi}$	$\frac{\pi r}{2}$
Semicircular arc	The points on the circle $\,\!x^2 + y^2 = r^2$ and above the $\,\!x$ axis	$\,\!0$	$\frac{2r}{\pi}$	$\,\!\pi r$
Arc of circle	The points on the curve (in polar coordinates) $\,\!r = \rho$, from $\,\!\theta = -\alpha$ to $\,\!\theta = \alpha$	$\frac{\rho\sin(\alpha)}{\alpha}$	$\,\!0$	$\,\!2\alpha \rho$

External links

• http://www.engineering.com/Library/ArticlesPage/tabid/85/articleType/ArticleView/articleId/109/Centroids-of-Common-Shapes.aspx
• http://www.efunda.com/math/areas/IndexArea.cfm