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 Image:Triangle_centroid_2.svg   \frac{h}{3} \frac{bh}{2}
Quarter-circular area Image:Quarter_circle_centroid.svg \frac{4r}{3\pi} \frac{4r}{3\pi} \frac{\pi r^2}{4}
Semicircular area Image:Semicircle_centroid.svg \,\!0 \frac{4r}{3\pi} \frac{\pi r^2}{2}
Quarter-elliptical area Image:Elliptical_quarter.svg \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

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