# Dictionary Definition

centroid n : the center of mass of an object of uniform density

# User Contributed Dictionary

## English

### Noun

1. In the context of "mathematics|physics": The point at the centre of any shape, sometimes called centre of area or centre of volume. For a triangle, the centroid is the point at which the medians intersect. The co-ordinates of the centroid are the average (arithmetic mean) of the co-ordinates of all the points of the shape. For a shape of uniform density, the centroid coincides with the centre of mass which is also the centre of gravity in a uniform gravitational field.

# Extensive Definition

|} 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 perpendicular distance between each side and the opposing point. (As illustrated in the figures to the right).
The centroid is the triangle's center of mass if the triangle is made from a uniform sheet of material. Its Cartesian coordinates are the means of the coordinates of the three vertices. That is, if the three vertices are located at (x_a, y_a), (x_b, y_b), and (x_c, y_c), then the centroid is at:
\Big(
\begin\frac13\end (x_a+x_b+x_c),\; \begin\frac13\end (y_a+y_b+y_c) \Big) = \begin\frac13\end (x_a, y_a) + \begin\frac13\end (x_b, y_b) + \begin\frac13\end (x_c, y_c).
A similar result holds for a tetrahedron: its centroid is the intersection of all line segments that connect each vertex to the centroid of the opposite face. These line segments are divided by the centroid in the ratio 3:1. The result generalizes to any n-dimensional simplex in the obvious way. If the set of vertices of a simplex is , then considering the vertices as vectors, the centroid is at:
\frac\sum_^n v_i
The isogonal conjugate of a triangle's centroid is its symmedian point.

### Proof that the centroid of a triangle divides each median in the ratio 2:1

Let the medians AD, BE and CF of the triangle ABC intersect at G, the centroid of the triangle, and let the straight line AD be extended up to the point O such that
AG = GO. \,
Then the triangles AGE and AOC are similar (common angle at A, AO is twice AG, AC is twice AE), and so OC is parallel to GE. But GE is BG extended, and so OC is parallel to BG. Similarly, OB is parallel to CG.
The figure GBOC is therefore a parallelogram. Since the diagonals of a parallelogram bisect one another, the point of intersection D between the diagonals GO and BC is such that GD = DO, and
GO = GD + DO = 2GD. \,
So, AG = GO = 2GD, \,
or AG:GD = 2:1. \,
This is true for every other median.

## Centroid of polygon

The centroid of a non-overlapping closed polygon defined by N vertices ( xi , yi ) can be calculated as follows. The notional vertex ( xN , yN ) is the same as ( x0 , y0 ).
The area of the polygon is given by:
A = \frac\sum_^ (x_i\ y_ - x_\ y_i)
The centroid of the polygon is then given by:
C_x = \frac\sum_^(x_i+x_)(x_i\ y_ - x_\ y_i)
C_y = \frac\sum_^(y_i+y_)(x_i\ y_ - x_\ y_i)

## Centroid of a finite set of points

Given a finite set of points x_1,x_2,\ldots,x_k in \mathbb^n, their centroid C is defined to be
C = \frac.

## Area centroid

The centroid of an area is very similar to the center of mass of a body. This is calculated using only the geometry of the figure. If the body is homogeneous, the center of mass will be at the centroid.
For a two body figure, you may have an equation that looks like this:
\overline = \dfrac
\overline is the distance from your reference coordinate axis to the centroid of the particular area. A is the area of that particular section.
The general function for calculating the centroid of a geometrically complex cross section is most easily applied when the figure is divided into known simple geometries and then applying the formula:
\overline = \frac
\overline = \frac
The distance from the y-axis to the centroid is \overline. The distance from the x-axis to the centroid is \overline. The coordinates of the centroid are (\overline , \overline).

## Integral formula

The abscissa (x coordinate) of the centroid of a plane figure can be given as the integral
C_x = \frac,
where f(x) is the extent of the object along the y axis at abscissa x, that is the measure of the figure's section at x. This formula can be derived from the first moment about the y axis of the area.
This process is equivalent to taking a weighted average. Supposing that the y axis represents frequency, and the x axis represents the variable whose average we want to find, then the location of the centroid along the x axis is simply the mean: \bar
Hence the centroid can be thought of as a weighted average of many infintesimally small elements that represent a particular shape.
The same formula yields the first coordinate of the centroid of an object in \R^n, for any dimension n, provided that f(x) is the (n-1)-dimensional measure of the object's cross-section at coordinate x — that is, the set of all points in the object whose first coordinate is x.
Note that the denominator is simply the object's n-dimensional measure. In the special case where f is normalized, i.e., the denominator is 1, the centroid is called the mean of f.
The formula cannot be applied if the object has zero measure, or if either integral diverges.

## Centroid of cone and pyramid

The centroid of a cone or pyramid is located on the line segment that connects the apex to the centroid of the base, and divides that segment in the ratio 3:1.

## Center of symmetry

If the centroid is defined, it is a fixed point of all isometries in its symmetry group. Thus symmetry may fully or partially determine the centroid, depending on the kind of symmetry. It also follows that for an object with translational symmetry the centroid is undefined, because a translation has no fixed point.

## References

centroid in Bulgarian: Медицентър
centroid in German: Schwerpunkt
centroid in Spanish: Centroide
centroid in French: Barycentre (géométrie affine)
centroid in Italian: Baricentro (geometria)
centroid in Hungarian: Súlypont
centroid in Dutch: Zwaartepunt
centroid in Polish: Centroid
centroid in Russian: Центроид