# Centroid

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The centroid of a body is the center of its mass (or masses), the point at which it would be stable, or balance, under the influence of gravity. There are other names for the same point. It is also often called the center of gravity and the geocenter and barycenter. The term is essentially the same as the Greek term known to Archimedes, kentron toi Bareos

There are three common "centers of gravity" that are studied in math, science and engineering. The most common in math is the center of masses located at the vertices of a polygon. This is more common because the other two cases can be reduced to a variation of this approach. It is this case of point masses at the vertices that I mean when I use centroid or center of gravity in this note, unless otherwise stated. A second approach is to treat the area of the polygon as if it were a sheet of uiniform density. The third, and least common, approach is to represent the sides of the polygon as wire rods of uniform density.

Most students are first introduced to the terms above in reference to a point in a triangle. Since the center of masses at the vertices in a triangle give the same point as a uniform sheet, they are often confused about the various distinctions. The three centers of gravity are usually different points in other non-symmetric polygons. It is this point, the center of balance for the uniform sheet and also of point masses at the vertices, that is almost universally referenced as the centroid of a triangle.
The centroid of a triangle is a point at the intersection of the three medians of the triangle.   One of the basic ideas known about the centroid is that it it divides the medians into a 2:1 ratio.  The part of the median nearest the vertex is always twice as long as the part near the midpoint of the side.   If the coordinates of the triangle are known, then the coordinates of the centroid are the averages of the coordinates of the vertices.  If we call the three vertices A=(x1,y1); B=(x2,y2) and C=(x3,y3) then the coordinates of the geocenter would be
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This is extendable to the centroid of a tetrahedron in three space.  If we construct the centroids of each of the triangular faces and construct the "medial segment" from each vertex to the centroid of the opposite face, they will also intersect in a single point, the centroid of the tetrahedron.  and the x,y,z coordinates of the centroid is the average of the corresponding coordinates of the four vertices.  The centroid of a tetrahedron divides the medial segments into a 3:1 ratio.

If the midpoints of opposite pairs of edges of a tetrahedron are connected they will all intersect at the centroid also. Recently while rereading Great Moments in Mathematics before 1650 by Howard Eves I found that the extension above was called "Commandino's theorem". It is named for
Federigo Commandino and was published in his De Centro Gravitatus Solidorum [On centers of gravity of solids] in 1565. Commandino translated many of the classic Greek mathematical texts and a quote from St Andrews web site says that Commandino "had the greatest influence of anybody in ensuring that the classic Greek mathematical texts survived by publishing his editions of them."

In a quadrilateral, the line joining the midpoints of two opposite sides is called a bimedian.  The centroid of masses located at the vertices of a quadrilateral is also the intersection of the bimedians of a quadrilateral.  Another property of the quadrilaterals centroid is that it is also the midpoint of the segment joining the midpoints of the diagonals.

The advantage of using the point mass approach to finding centers of gravity is that the other two common cases can be reduced to point masses of uneven weights very easily. To find centers of gravity of uniform density sheets, one can simply divide the polygon into non-overlapping triangles and treat the system as a set of point masses at the centroids of these triangles with a mass equal to the area of the triangle. To find the center of uniform rods along the perimeter of a polygon, replace each side with a point mass equal to the length of the line located at its midpoint. The center of gravity of uniform wire rods on the perimeter of a triangle is the Spieker point, which is the incenter of the medial triangle. Professor Kimberling has a page showing how to find the center of mass of any shape by a physical method.

A direct method of finding the center of gravity of a uniform density sheet in the shape of a quadrilateral was found by F. Wittenbauer (1857-1922). If the triesectors of each edge of the quadrilateral are found, and lines are drawn through each pair of trisectors adjacent to a vertex, they form a parallelogram, Wittenbauer's Parallelogram. The center of the parallelogram is the center of gravity of the uniform sheet.

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The center of gravity of the boundary of a triangle If the sides of a triangle are treated as rods, as above, the problem of finding the center of gravity can be reduced to the case of points of mass by assigning a point of mass to the midpoint of each side with a mass value equal to its length. The center of mass coordinates for x and y can then each be computed independently. Multiply the coordinate values times the mass value and find the sum of these; then divide by the total mass (the perimeter of the triangle).

As an example, a 3,4,5 right triangle with the right angle vertex at the oring would have midpoints at (1.5,0) with a mass of three; (0,2) with a mass of four; and (1.5,2) with a mass of five. The x-coordinate of the center of mass would be found by ((1.5)(3)+0(4)+1.5(5))/12 which turns out nicely to be x=1. The Y-coordinate can be found by ((0)(3)+2(4)+2(5))/12, giving a y-coordinate of 3/2. Thus the centroid of the edges of the triangle is at (1,3/2). Using a method known as early as Archimedes, you can reduce this to the geometric construction of finding the point of intersection of the angle bisectors of the medial triangle.

The word is based on the word center and the Greek suffix oid and means "center like".  It probably is a relatively modern word, perhaps created after 1850.