The centroid is center of mass of a two-dimensional planar lamina or a three-dimensional solid. The mass of a lamina with surface density function is
(1)
and the coordinates of the centroid (also called the center of gravity) are
The centroid of a lamina is the point on which it would balance when placed on a needle. The centroid of a solid is the point on which the solid would "balance."
The geometric centroid of a region can be computed in the Wolfram Language using RegionCentroid[reg].
The centroid of a set of point masses located at positions is
(4)
which, if all masses are equal, simplifies to
(5)
For a closed lamina of uniform density with boundary specified by for and the lamina on the left as the curve is traversed, Green's theorem can be used to compute the centroid as
The positions of the geometric centroid of a planar non-self-intersecting polygon with vertices , ..., are
where is the polygon area and and (Bourke 1988, Nürnberg 2013).
The centroid of the vertices of a quadrilateral occurs at the point of intersection of the bimedians (i.e., the lines and joining pairs of opposite midpoints) (Honsberger 1995, pp. 36-37). In addition, it is the midpoint of the line connecting the midpoints of the diagonals and (Honsberger 1995, pp. 39-40).
Given an arbitrary hexagon, connecting the centroids of each consecutive three sides gives the so-called centroid hexagon, a hexagon with equal and parallel sides (Wells 1991).
The centroid of a semicircle of radius is given by
(10)
The centroids of several common laminas bounded by the following curves along the nonsymmetrical axis are summarized in the following table.
In three dimensions, the mass of a solid with density function is
(11)
and the coordinates of the center of mass are
See alsoCentroid Hexagon,
Circumcenter of Mass,
Pappus's Centroid Theorem,
Polygon Centroid,
Polyhedron Centroid Explore with Wolfram|Alpha ReferencesBeyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 132, 1987.Bourke, P. "Calculating the Area and Centroid of a Polygon." July 1988. http://paulbourke.net/geometry/polygonmesh/.Honsberger, R. Episodes in Nineteenth and Twentieth Century Euclidean Geometry. Washington, DC: Math. Assoc. Amer., 1995.Kern, W. F. and Bland, J. R. "Center of Gravity." §39 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, p. 110, 1948.McLean, W. G. and Nelson, E. W. "First Moments and Centroids." Ch. 9 in Schaum's Outline of Theory and Problems of Engineering Mechanics: Statics and Dynamics, 4th ed. New York: McGraw-Hill, pp. 134-162, 1988.Nürnberg, R. "Calculating the Area and Centroid of a Polygon in 2D." 2013. https://www.ma.imperial.ac.uk/~rn/centroid.pdf.Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, 1999.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 53-54, 1991. Referenced on Wolfram|AlphaGeometric Centroid Cite this as:Weisstein, Eric W. "Geometric Centroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeometricCentroid.html
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