The geometric centroid of a region is the arithmetic mean position of its points. It depends only on the geometry of the region and coincides with its center of mass when the density is uniform. For a two-dimensional planar lamina of area
 |
(1)
|
the coordinates of the geometric centroid are
 |  |  |
(2)
|
 |  |  |
(3)
|
The centroid of a uniform-density lamina is the point on which it would balance when placed on a needle. Similarly, the centroid of a uniform-density 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 
points located at positions 
is
 |
(4)
|
For a closed lamina of uniform density with boundary specified by 
for ![t in [t_0,t_1]](/images/equations/GeometricCentroid/Inline10.svg)
and the lamina on the left as the curve is traversed,
Green's theorem can be used to compute the centroid
as
 |  |  |
(5)
|
 |  |  |
(6)
|
The positions of the geometric centroid of a planar non-self-intersecting polygon with vertices 
, ..., 
are
 |  |  |
(7)
|
 |  |  |
(8)
|
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
 |
(9)
|
The centroids of several common laminas bounded by the following curves along the nonsymmetrical axis are summarized in the following table.
In three dimensions, a solid has volume
 |
(10)
|
and the coordinates of its geometric centroid are
 |  |  |
(11)
|
 |  |  |
(12)
|
 |  |  |
(13)
|
, 
, 
