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
 |  |  |
(2)
|
 |  |  |
(3)
|
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 ![t in [t_0,t_1]](/images/equations/GeometricCentroid/Inline12.svg)
and the lamina on the left as the curve is traversed,
Green's theorem can be used to compute the centroid
as
 |  |  |
(6)
|
 |  |  |
(7)
|
The positions of the geometric centroid of a planar non-self-intersecting polygon with vertices 
, ..., 
are
 |  |  |
(8)
|
 |  |  |
(9)
|
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
 |  |  |
(12)
|
 |  |  |
(13)
|
 |  |  |
(14)
|
, 
, 
