RegionCentroid [reg]
gives the centroid of the region reg.
RegionCentroid
RegionCentroid [reg]
gives the centroid of the region reg.
Details and Options
- RegionCentroid is also known as center of mass, center of gravity, or barycenter.
- The centroid is effectively given by Integrate [{x1,…,xn},{x1,…,xn}∈reg]/RegionMeasure [reg].
- The centroid is in the region when the region is convex. Otherwise it is typically not in the region.
- Examples of cases where rows correspond to embedding dimension and columns to geometric dimension:
- If the region reg consists of a finite number of points, the RegionCentroid gives the mean.
- A region with infinite RegionMeasure has no RegionCentroid and returns a point with Indeterminate coordinates.
- RegionCentroid takes an Assumptions option that can be used to specify assumptions on parameters.
- RegionCentroid can be used with symbolic regions in GeometricScene .
Examples
open all close allBasic Examples (2)
Find the centroid of a region:
The centroid of a Polygon :
Scope (21)
Special Regions (10)
The centroid for Point corresponds to the mean of the coordinates:
Points can be used in any number of dimensions:
Line :
Lines can be used in any number of dimensions:
Rectangle can be used in 2D:
Cuboid can be used in any number of dimensions:
A 4D Cuboid :
A Simplex can correspond to a point, line, or triangle in 2D:
Simplices can be used in any number of dimensions:
The centroid of a standard unit Simplex in dimension :
The centroid of a Polygon may lie outside the region:
In 3D:
Disk can be used in 2D:
Ball can be used in any dimension:
In 4D:
Disk as an ellipse can be used in 2D:
Ellipsoid can be used in any dimension:
Circle can be used in 2D:
As an ellipse:
Cylinder can be used in 3D:
Cone can be used in 3D:
Formula Regions (2)
The centroid of a disk represented as an ImplicitRegion :
The centroid of a cylinder:
The centroid of a disk represented as a ParametricRegion :
Using a rational parameterization of the disk:
The centroid of a cylinder:
Mesh Regions (2)
The centroid of a MeshRegion :
A 1D mesh embedded in 2D:
In 3D:
The centroid of a BoundaryMeshRegion :
In 3D:
Derived Regions (5)
The centroid of a RegionIntersection :
The centroid of a TransformedRegion :
The centroid of a RegionBoundary :
General Boolean combination :
Inverse transformed region :
Geographic Regions (2)
The centroid of a polygon with GeoPosition :
The centroid of a polygon with GeoGridPosition :
Applications (5)
Find the center of mass for a mesh region:
Compute the center of mass of a region with density given by and compare to the centroid:
Visualize it:
The center of mass is shifted because the density is highest in the lower-left:
Find a perpendicular bisector of a triangle:
Visualize circumcenter and bisectors in red:
Compute a centroidal Voronoi diagram from a random set of points:
Define a function that computes the centroid of each Voronoi region in a Voronoi mesh:
Recursively apply VoronoiMesh to the centroids of the precursive Voronoi regions:
Visualize the Voronoi mesh and centroids at each iteration:
The generating point (black) of each Voronoi region converges toward the region's centroid (red):
Estimate the centroid of a region by taking the Mean of a random sampling of points in the region:
Properties & Relations (3)
RegionCentroid is not necessarily in the region if the region is not convex:
RegionCentroid is equivalent to Integrate [p,p∈ℛ]/m with m=RegionMeasure [ℛ]:
The centroid for is given by when disjoint:
Possible Issues (1)
RegionCentroid returns a point with Indeterminate coordinates for a region with infinite RegionMeasure :
See Also
RegionMeasure Integrate MomentOfInertia RegionMoment Mean Expectation GeometricScene
Function Repository: BarycentricCoordinates
Related Guides
History
Text
Wolfram Research (2014), RegionCentroid, Wolfram Language function, https://reference.wolfram.com/language/ref/RegionCentroid.html.
CMS
Wolfram Language. 2014. "RegionCentroid." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RegionCentroid.html.
APA
Wolfram Language. (2014). RegionCentroid. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RegionCentroid.html
BibTeX
@misc{reference.wolfram_2025_regioncentroid, author="Wolfram Research", title="{RegionCentroid}", year="2014", howpublished="\url{https://reference.wolfram.com/language/ref/RegionCentroid.html}", note=[Accessed: 04-January-2026]}
BibLaTeX
@online{reference.wolfram_2025_regioncentroid, organization={Wolfram Research}, title={RegionCentroid}, year={2014}, url={https://reference.wolfram.com/language/ref/RegionCentroid.html}, note=[Accessed: 04-January-2026]}