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, 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.
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 |
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-axe | 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-axe | 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-axe | 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-axe | a, b = the principal semi-axes of the base ellipse c = the principal z-semi-axe from the center of base ellipse |
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