Centroid

Informally, it can be understood as the point at which a cutout of the shape (with uniformly distributed mass) could be perfectly balanced on the tip of a pin.

[citation needed] The center of gravity, as the name indicates, is a notion that arose in mechanics, most likely in connection with building activities.

It is uncertain when the idea first appeared, as the concept likely occurred to many people individually with minor differences.

Nonetheless, the center of gravity of figures was studied extensively in Antiquity; Bossut credits Archimedes (287–212 BCE) with being the first to find the centroid of plane figures, although he never defines it.

[5] It is unlikely that Archimedes 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 the Elements.

The first explicit statement of this proposition is due to Heron of Alexandria (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.

The centroid of a ring or a bowl, for example, lies in the object's central void.

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 hyperplanes of symmetry.

The centroid of many figures (regular polygon, regular polyhedron, cylinder, rectangle, rhombus, circle, sphere, ellipse, ellipsoid, superellipse, superellipsoid, etc.)

The centroid of a uniformly dense planar lamina, such as in figure (a) below, may be determined experimentally by using a plumbline and a pin to find the collocated center of mass of a thin body of uniform density having the same shape.

The unique intersection point of these lines will be the centroid (figure c).

This method can be extended (in theory) to concave shapes where the centroid may lie outside the shape, and virtually to solids (again, of uniform density), where the centroid may lie within the body.

The (virtual) positions of the plumb lines need to be recorded by means other than by drawing them along the shape.

In principle, progressively narrower cylinders can be used to find the centroid to arbitrary precision.

However, by marking the overlap range from multiple balances, one can achieve a considerable level of accuracy.

The horizontal position of the centroid, from the left edge of the figure is

The mathematical principle involved is a special case of Green's theorem.

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).

In trilinear coordinates the centroid can be expressed in any of these equivalent ways in terms of the side lengths

[15] A triangle's centroid lies on its Euler line between its orthocenter

This is not true for other lines through the centroid; the greatest departure from the equal-area 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

In these formulae, the vertices are assumed to be numbered in order of their occurrence along the polygon's perimeter; furthermore, the vertex

computed as above, will be negative; however, the centroid coordinates will be correct even in this case.)

For a cone or pyramid that is just a shell (hollow) with no base, the centroid is

A line segment joining a vertex of a tetrahedron with the centroid of the opposite face is called a median, and a line segment joining the midpoints of two opposite edges is called a bimedian.

The centroid of a tetrahedron is the midpoint between its Monge point and circumcenter (center of the circumscribed sphere).

The centroid of a solid hemisphere (i.e. half of a solid ball) divides the line segment connecting the sphere's center to the hemisphere's pole in the ratio

The centroid of a hollow hemisphere (i.e. half of a hollow sphere) divides the line segment connecting the sphere's center to the hemisphere's pole in half.