In mathematics and physics, the centroid or geometric center of a plane figure is the arithmetic mean position of all the points in the figure. Informally, it is the point at which a cutout of the shape could be perfectly balanced on the tip of a pin.
The definition extends to any object in n space: its centroid is the mean position of all the points in all of the coordinate directions.
While in geometry the term barycenter is a synonym for centroid, in astrophysics and astronomy, barycenter is the center of mass of two or more bodies that orbit each other. In physics, the center of mass is the arithmetic mean of all points Weighting by the local density or specific weight. If a physical object has uniform density, then its center of mass is the same as the centroid of its shape.
In geography, the centroid of a radial projection of a region of the Earth's surface to sea level is known as the region's geographical center.
The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities. When, where, and by whom it was invented is not known, as it is a concept that likely occurred to many people individually with minor differences.
While it is possible Euclid was still active in Alexandria during the childhood of Archimedes (287212 BCE), it is certain that when Archimedes visited Alexandria, Euclid was no longer there. Thus Archimedes could not have learned the theorem that the medians of a triangle meet in a point—the center of gravity of the triangle directly from Euclid, as this proposition is not in Euclid's Elements. The first explicit statement of this proposition is due to Heron (perhaps the first century CE) and occurs in his Mechanics. It may be added, in passing, that the proposition did not become common in the textbooks on plane geometry until the nineteenth century.
While Archimedes does not state that proposition explicitly, he makes indirect references to it, suggesting he was familiar with it. However, Jean Etienne Montucla (17251799), the author of the first history of mathematics (1758), declares categorically (vol. I, p. 463) that the center of gravity of solids is a subject Archimedes did not touch.
In 1802 Charles Bossut (17301813) published a twovolume Essai aur PhisMire generale des mathematiques. This book was highly esteemed by his contemporaries, judging from the fact that within two years after its publication it was already available in translation in Italian (180203), English (1803), and German (1804). Bossut credits Archimedes with having found the centroid of plane figures, but has nothing to say about solids.
If the centroid is defined, it is a fixed point of all isometries in its symmetry group. In particular, the geometric centroid of an object lies in the intersection of all its of symmetry. The centroid of many figures (regular polygon, regular polyhedron, cylinder, rectangle, rhombus, circle, sphere, ellipse, ellipsoid, superellipse, superellipsoid, etc.) can be determined by this principle alone.
In particular, the centroid of a parallelogram is the meeting point of its two . This is not true for other .
For the same reason, the centroid of an object with translational symmetry is undefined (or lies outside the enclosing space), because a translation has no fixed point.
Any of the three medians through the centroid divides the triangle's area in half. This is not true for other lines through the centroid; the greatest departure from the equalarea division occurs when a line through the centroid is parallel to a side of the triangle, creating a smaller triangle and a trapezoid; in this case the trapezoid's area is 5/9 that of the original triangle.
Let P be any point in the plane of a triangle with vertices A, B, and C and centroid G. Then the sum of the squared distances of P from the three vertices exceeds the sum of the squared distances of the centroid G from the vertices by three times the squared distance between P and G:
The sum of the squares of the triangle's sides equals three times the sum of the squared distances of the centroid from the vertices:
A triangle's centroid is the point that maximizes the product of the directed distances of a point from the triangle's sidelines.Clark Kimberling, "Trilinear distance inequalities for the symmedian point, the centroid, and other triangle centers", Forum Geometricorum, 10 (2010), 135139. http://forumgeom.fau.edu/FG2010volume10/FG201015index.html
For other properties of a triangle's centroid, see below.
(a)  (b)  (c) 
This method can be extended (in theory) to concave shapes where the centroid lies outside the shape, and to solids (of uniform density), but the positions of the plumb lines need to be recorded by means other than drawing.
Holes in the figure $X$, overlaps between the parts, or parts that extend outside the figure can all be handled using negative areas $A\_i$. Namely, the measures $A\_i$ should be taken with positive and negative signs in such a way that the sum of the signs of $A\_i$ for all parts that enclose a given point $p$ is 1 if $p$ belongs to $X$, and 0 otherwise.
For example, the figure below (a) is easily divided into a square and a triangle, both with positive area; and a circular hole, with negative area (b).
The centroid of each part can be found in any list of centroids of simple shapes (c). Then the centroid of the figure is the weighted average of the three points. The horizontal position of the centroid, from the left edge of the figure is
The same formula holds for any threedimensional objects, except that each $A\_i$ should be the volume of $X\_i$, rather than its area. It also holds for any subset of $\backslash R^d$, for any dimension $d$, with the areas replaced by the $d$dimensional measures of the parts.
where the integrals are taken over the whole space $\backslash R^n$, and g is the characteristic function of the subset, which is 1 inside X and 0 outside it. Note that the denominator is simply the measure of the set X. This formula cannot be applied if the set X has zero measure, or if either integral diverges.
Another formula for the centroid is
where C_{ k} is the kth coordinate of C, and S_{ k}( z) is the measure of the intersection of X with the hyperplane defined by the equation x_{ k} = z. Again, the denominator is simply the measure of X.
For a plane figure, in particular, the barycenter coordinates are
where A is the area of the figure X; S_{y}( x) is the length of the intersection of X with the vertical line at abscissa x; and S_{x}( y) is the analogous quantity for the swapped axes.
where $A$ is the area of the region (given by $\backslash int\_a^b\; f(x)\backslash ;dx$).
The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1, which is to say it is located ⅓ of the distance from each side to the opposite vertex (see figures at right). Its Cartesian coordinates are the arithmetic mean of the coordinates of the three vertices. That is, if the three vertices are $L\; =\; (x\_L,\; y\_L),$ $M=\; (x\_M,\; y\_M),$ and $N=\; (x\_N,\; y\_N),$ then the centroid (denoted C here but most commonly denoted G in triangle geometry) is
C = \frac13(L+M+N) = \left(\frac13 (x_L+x_M+x_N),\;\; \frac13(y_L+y_M+y_N)\right).
The centroid is therefore at $\backslash tfrac13:\backslash tfrac13:\backslash tfrac13$ in barycentric coordinates.
In trilinear coordinates the centroid can be expressed in any of these equivalent ways in terms of the side lengths a, b, c and vertex angles L, M, N:Clark Kimberling's Encyclopedia of Triangles
The centroid is also the physical center of mass if the triangle is made from a uniform sheet of material; or if all the mass is concentrated at the three vertices, and evenly divided among them. On the other hand, if the mass is distributed along the triangle's perimeter, with uniform linear density, then the center of mass lies at the Spieker center (the incenter of the medial triangle), which does not (in general) coincide with the geometric centroid of the full triangle.
The area of the triangle is 1.5 times the length of any side times the perpendicular distance from the side to the centroid.
A triangle's centroid lies on its Euler line between its orthocenter H and its circumcenter O, exactly twice as close to the latter as to the former:
In addition, for the incenter I and ninepoint center N, we have
If G is the centroid of the triangle ABC, then:
The isogonal conjugate of a triangle's centroid is its symmedian.
and where A is the polygon's signed area, as described by
In these formulas, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex ( x_{ n}, y_{ n} ) is assumed to be the same as ( x_{0}, y_{0} ), meaning i + 1 on the last case must loop around to i = 0. (If the points are numbered in clockwise order, the area A, computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)
These results generalize to any ndimensional simplex in the following way. If the set of vertices of a simplex is $\{v\_0,\backslash ldots,v\_n\}$, then considering the vertices as vectors, the centroid is
The geometric centroid coincides with the center of mass if the mass is uniformly distributed over the whole simplex, or concentrated at the vertices as n equal masses.

