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

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For each two-dimensional shape below, the area and the centroid coordinates   are given:

Shape Figure     Area
rectangle area        
General triangular area    [1]    
Isosceles-triangular area        
Right-triangular area        
Circular area        
Quarter-circular area[2]        
Semicircular area[3]        
Circular sector        
Circular segment        
Annular sector        
Quarter-circular arc The points on the circle   and in the first quadrant      
Semicircular arc The points on the circle   and above the   axis      
Arc of circle The points on the curve (in polar coordinates)  , from   to        
elliptical area        
Quarter-elliptical area        
Semielliptical area        
Parabolic area The area between the curve   and the line        
Semiparabolic area

The area between the curve   and the   axis, from   to  

       
Parabolic spandrel The area between the curve   and the   axis, from   to        
General spandrel The 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

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For each three-dimensional body below, the volume and the centroid coordinates   are given:

Shape Figure       Volume
Cuboid a, b = the sides of the cuboid's base
c = the third side of the cuboid
       
Right-rectangular pyramid a, b = the sides of the base
h = the distance is from base to the apex
       
General triangular prism b = the base side of the prism's triangular base,
h = the height of the prism's triangular base
L = the length of the prism
see above
for general
triangular base
     
Isosceles triangular prism b = the base side of the prism's triangular base,
h = the height of the prism's triangular base
L = the length of the prism
       
Right-triangular prism b = the base side of the prism's triangular base,
h = the perpendicular side of the prism's triangular base
L = the length of the prism
       
Right circular cylinder r = the radius of the cylinder
h = the height of the cylinder
       
Right circular solid cone r = the radius of the cone's base
h = the distance is from base to the apex
       
Solid sphere r = the radius of the sphere        
Solid hemisphere r = the radius of the hemisphere        
Solid 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 edge
       
Solid 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 edge
       
Solid 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 ellipse
       
Solid 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

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References

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  1. ^ "Coordinates of a triangle centroid with calculator (Coordinate Geometry) - Math Open Reference".
  2. ^ "Quarter Circle". eFunda. Retrieved 23 April 2016.
  3. ^ "Circular Half". eFunda. Retrieved 23 April 2016.
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