Geometric Centroid (original) (raw)

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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 sigma(x,y) is

M=intintsigma(x,y)dA, (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 n point masses m_i located at positions x_i is

x^_=(sum_(i=1)^(n)m_ix_i)/(sum_(i=1)^(n)m_i), (4)

which, if all masses are equal, simplifies to

x^_=(sum_(i=1)^(n)x_i)/n. (5)

For a closed lamina of uniform density with boundary specified by (x(t),y(t)) for t in [t_0,t_1] 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 (x_1,y_1), ..., (x_n,y_n) are

where A is the polygon area and x_(n+1)=x_1 and y_(n+1)=y_1 (Bourke 1988, Nürnberg 2013).

QuadrilateralCentroid

The centroid of the vertices of a quadrilateral occurs at the point of intersection of the bimedians (i.e., the lines M_(AB)M_(CD) and M_(AD)M_(BC) joining pairs of opposite midpoints) (Honsberger 1995, pp. 36-37). In addition, it is the midpoint of the line M_(AC)M_(BD) connecting the midpoints of the diagonals AC and BD (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 R is given by

x^_=(2R)/pi. (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 rho(x,y,z) is

M=intintintrho(x,y,z)dV, (11)

and the coordinates of the center of mass are

See also

Centroid Hexagon, Circumcenter of Mass, Pappus's Centroid Theorem,Polygon Centroid, Polyhedron Centroid

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References

Beyer, 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.

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Geometric Centroid

Cite this as:

Weisstein, Eric W. "Geometric Centroid." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GeometricCentroid.html

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