Centroid of a triangle

In geometry, the centroid or barycenter of an object X in n-dimensional space is the intersection of all hyperplanes that divide X into two parts of equal moment about the hyperplane. Informally, it is the "average" of all points of X. Image File history File links Triangle. ... Image File history File links Triangle. ... For other uses, see Geometry (disambiguation). ... 2-dimensional renderings (ie. ... Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... A hyperplane is a concept in geometry. ... -1... Averages redirects here. ...

The centroid of an object coincides with its center of mass if the object has uniform density, or if the object's shape and density have a symmetry which fully determines the centroid. These conditions are sufficient but not necessary. In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... For other uses, see Density (disambiguation). ...

The centroid of a finite set of points can be computed as the arithmetic mean of each coordinate of the points. In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

In geography, the centroid of a region of the Earth's surface is known as its geographical center.

The centroid of a convex object always lies in the object. A non-convex object might have a centroid that is outside the figure itself. The centroid of a ring or a bowl, for example, lies in the object's central void. Look up Convex set in Wiktionary, the free dictionary. ... An annulus In mathematics, an annulus (the Latin word for little ring, with plural annuli) is a ring-shaped geometric figure, or more generally, a term used to name a ring-shaped object. ... Look up Bowl in Wiktionary, the free dictionary. ...

Centroid of triangle and tetrahedron

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). For alternate meanings, such as the musical instrument, see triangle (disambiguation). ... The triangle medians and the centroid. ... In geometry, a vertex (plural vertices) is a special kind of point, usually a corner of a polygon, polyhedron, or higher dimensional polytope. ... This article is about the mathematical concept. ...

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: In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ... Cartesian means relating to the French mathematician and philosopher Descartes, who, among other things, worked to merge algebra and Euclidean geometry. ... In mathematics and statistics, the arithmetic mean (or simply the mean) of a list of numbers is the sum of all the members of the list divided by the number of items in the list. ...

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 v₀,...,v_n, then considering the vertices as vectors, the centroid is at: For the academic journal, see Tetrahedron (journal). ... A 3-simplex or tetrahedron In geometry, a simplex (plural simplexes or simplices) or n-simplex is an n-dimensional analogue of a triangle. ... This article is about vectors that have a particular relation to the spatial coordinates. ...

The isogonal conjugate of a triangle's centroid is its symmedian point. In geometry, the isogonal conjugate of a point P with respect to a triangle ABC is constructed by reflecting the lines PA, PB, and PC about the angle bisectors of A, B, and C. These three reflected lines concur at the isogonal conjugate of P. The isogonal conjugate of a... In geometry, three special lines are associated with every triangle, the triangles symmedians. ...

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

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. // Two geometrical objects are called similar if one is congruent to the result of a uniform scaling (enlarging or shrinking) of the other. ...

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 A parallelogram. ... For the bisection theorem, see ham sandwich theorem. ...

So,

or

This is true for every other median.

Centroid of polygon

The centroid of a non-overlapping closed polygon defined by N vertices ( x_i , y_i ) can be calculated as follows.^[1] The notional vertex ( x_N , y_N ) is the same as ( x₀ , y₀ ).

The area of the polygon is given by:

The centroid of the polygon is then given by:

Centroid of a finite set of points

Given a finite set of points in , their centroid C is defined to be

.

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.^[2] In physics, the center of mass of a system of particles is a specific point at which, for many purposes, the systems mass behaves as if it were concentrated. ...

For a two body figure, you may have an equation that looks like this:

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:

The distance from the y-axis to the centroid is . The distance from the x-axis to the centroid is . The coordinates of the centroid are .

Integral formula

The abscissa (x coordinate) of the centroid of a plane figure can be given as the integral Abscissa means the x coordinate on an (x, y) graph; the input of a mathematical function against which the output is plotted. ...

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. The first moment of area, sometimes misnamed as the first moment of inertia, is based in the mathematical construct moments in metric spaces, stating that the moment of area equals the summation of area times distance to an axis [Σ(a x d)]. It is a measure of the distribution of...

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:

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 , 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. In mathematics the concept of a measure generalizes notions such as length, area, and volume (but not all of its applications have to do with physical sizes). ... This article is about mathematical mean. ...

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. Look up apex in Wiktionary, the free dictionary. ...

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. A fixed point of an isometry group is a point that is a fixed point for every isometry in the group. ... The symmetry group of an object (e. ... Sphere symmetry group o. ... In geometry, a translation slides an object by a vector a: Ta(p) = p + a. ...

See also

• List of centroids
• Pappus's centroid theorem

External links http://www. ... Pappuss centroid theorem consists of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. ...

References

1. ^ Calculating the area and centroid of a polygon
2. ^ Area Centroid

External links

• Encyclopedia of Triangle Centers by Clark Kimberling. The centroid is indexed as X(2).
• Triangle centers by Antonio Gutierrez from Geometry Step by Step from the Land of the Incas.
• Characteristic Property of Centroid at cut-the-knot
• Barycentric Coordinates at cut-the-knot
• Interactive animations showing Centroid of a triangle and Centroid construction with compass and straightedge