# List of centroids

The following diagrams show 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.

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 $\frac{2r}{\pi}$ $\frac{2r}{\pi}$ $L=\frac{\pi r}{2}$
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 $\,\!\theta = \alpha$ $\frac{\rho\sin(\alpha)}{\alpha}$ $\,\!0$ $L=\,\!2\alpha \rho$