List of centroids

The following is a list of centroids of various two-dimensional objects. 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. In the case of two-dimensional objects shown below, the hyperplanes are simply lines.

Centroids

Shape	Figure	$\bar x$	$\bar y$	Area
Right-triangular area		$\frac{b}{3}$	$\frac{h}{3}$	$\frac{bh}{2}$
Quarter-circular area[1]		$\frac{4r}{3\pi}$	$\frac{4r}{3\pi}$	$\frac{\pi r^2}{4}$
Semicircular area[2]		$\,\!0$	$\frac{4r}{3\pi}$	$\frac{\pi r^2}{2}$
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		$\frac{2r\sin(\alpha)}{3\alpha}$	$\,\!0$	$\,\!\alpha r^2$
Circular segment		$\frac{4r\sin^3(\alpha)}{3(2\alpha-\sin(2\alpha))}$	$\,\!0$	$\frac{r^2}{2}(2\alpha -\sin(2\alpha))$
Quarter-circular arc	The points on the circle $\,\!x^2 + y^2 = r^2$ and in the first quadrant	$\frac{2r}{\pi}$	$\frac{2r}{\pi}$	$L=\frac{\pi r}{2}$
Semicircular arc	The points on the circle $\,\!x^2 + y^2 = r^2$ and above the $\,\!x$ axis	$\,\!0$	$\frac{2r}{\pi}$	$L=\,\!\pi r$
Arc of circle	The points on the curve (in polar coordinates) $\,\!r = \rho$, from $\,\!\theta = -\alpha$ to $\,\!\theta = \alpha$	$\frac{\rho\sin(\alpha)}{\alpha}$	$\,\!0$	$L=\,\!2\alpha \rho$

See also

• List of moments of inertia
• List of second moments of area

References

1. "Quarter Circle". eFunda. Retrieved 23 April 2016.
2. "Circular Half". eFunda. Retrieved 23 April 2016.

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