List of centroids

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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 Triangle centroid 2.svg \frac{2b}{3} \frac{h}{3} \frac{bh}{2}
Quarter-circular area Quarter circle centroid.svg \frac{4r}{3\pi} \frac{4r}{3\pi} \frac{\pi r^2}{4}
Semicircular area Semicircle centroid2.svg \,\!0 \frac{4r}{3\pi} \frac{\pi r^2}{2}
Quarter-elliptical area Elliptical quarter.svg \frac{4a}{3\pi} \frac{4b}{3\pi} \frac{\pi a b}{4}
Semielliptical area Elliptical half.svg \,\!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 Circularsegment centroid.svg \,\!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[edit]