The following is a list of centroids of various two-dimensional and three-dimensional objects. The centroid of an object in -dimensional space is the intersection of all hyperplanes that divide into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of . For an object of uniform composition, or in other words, has the same density at all points, the centroid of a body is also its center of mass. In the case of two-dimensional objects shown below, the hyperplanes are simply lines.
2-D Centroids For each two-dimensional shape below, the area and the centroid coordinates are given: Shape Figure Arearectangle area General triangular area Isosceles-triangular area Right-triangular area Circular area Quarter-circular area Semicircular area Circular sector Circular segment Annular sector Quarter-circular arcThe points on the circle and in the first quadrantSemicircular arcThe points on the circle and above the axisArc of circleThe points on the curve (in polar coordinates) , from to elliptical area Quarter-elliptical area Semielliptical areaParabolic areaThe area between the curve and the line Semiparabolic area The area between the curve and the axis, from to Parabolic spandrelThe area between the curve and the axis, from to General spandrelThe area between the curve and the axis, from to
Where the centroid coordinates are marked as zero, the coordinates are at the origin, and the equations to get those points are the lengths of the included axes divided by two, in order to reach the center which in these cases are the origin and thus zero.
3-D Centroids For each three-dimensional body below, the volume and the centroid coordinates are given:
Shape Figure VolumeCuboida, b = the sides of the cuboid's base c = the third side of the cuboidRight-rectangular pyramida, b = the sides of the base h = the distance is from base to the apexGeneral triangular prismb = the base side of the prism's triangular base, h = the height of the prism's triangular base L = the length of the prismsee above for general triangular baseIsosceles triangular prismb = the base side of the prism's triangular base, h = the height of the prism's triangular base L = the length of the prismRight-triangular prismb = the base side of the prism's triangular base, h = the perpendicular side of the prism's triangular base L = the length of the prismRight circular cylinderr = the radius of the cylinder h = the height of the cylinderRight circular solid cone r = the radius of the cone's base h = the distance is from base to the apexSolid sphere r = the radius of the sphere Solid hemisphere r = the radius of the hemisphereSolid semi-ellipsoid of revolution around z-axis a = the radius of the base circle h = the height of the semi-ellipsoid from the base cicle's center to the edgeSolid paraboloid of revolution around z-axis a = the radius of the base circle h = the height of the paboloid from the base cicle's center to the edgeSolid ellipsoid a, b, c = the principal semi-axes of the ellipsoid Solid semi-ellipsoid around z-axis a, b = the principal semi-axes of the base ellipse c = the principal z-semi-axe from the center of base ellipseSolid paraboloid around z-axis a, b = the principal semi-axes of the base ellipse c = the principal z-semi-axe from the center of base ellipse
See also
List of moments of inertia List of second moments of area
References
External links http://www.engineering.com/Library/ArticlesPage/tabid/85/articleType/ArticleView/articleId/109/Centroids-of-Common-Shapes.aspx