They may be applied, for instance, to finding the centroid of a semicircle or of the arc of a semicircle.
The centroid of a hemisphere of radius R, for instance, is the same as the centroid of particles of masses 0, 7rR 2, and 4.
The centroid is at distance 8R from the plane face.
The ideas of moment and of centroid are extended to geometrical figures, whether solid, superficial or linear.
A line through the centroid of a plane figure (drawn in the plane of the figure) is a central line, and a plane through the centroid of a solid figure is a central plane, of the figure.
In the case, therefore, of any solid whose cross-section at distance x from one end is a quadratic function of x, the position of the crosssection through the centroid is to be found by determining the position of the centre of gravity of particles of masses proportional to So, S2, and 4S 1, placed at the extremities and the middle of a line drawn from one end of the solid to the other.
This point is the centroid of the body.
The centroid of a rectangle is its centre, i.e.
The centroid of a figure is a point fixed with regard to the figure, and such that its moment with regard to any plane (or, in the case of a plane area or line, with regard to any line in the plane) is the same as if the whole volume, area or length were concentrated at this point.
The centroid is sometimes called the centre of volume, centre of area, or centre of arc. The proof of the existence of the centroid of a figure is the same as the proof of the existence of the centre of gravity of a body.