List of centroids

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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 $X$ .

Shape	Figure	$\bar x$	$\bar y$	Area
Triangular area			$\frac{h}{3}$	$\frac{bh}{2}$
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	The area inside the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and above the $\,\!x$ axis	$\,\!0$	$\frac{4b}{3\pi}$	$\frac{\pi a b}{2}$
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 $\,\!\theta = \alpha$	$\frac{2\rho\sin(\alpha)}{3\alpha}$	$\,\!0$	$\,\!\alpha \rho^2$
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 $\,\!\theta = \alpha$	$\frac{\rho\sin(\alpha)}{\alpha}$	$\,\!0$	$\,\!2\alpha \rho$