The centroid of a geometric shape is the "average" of all points within the shape. That is, it is the point where all lines that divide the shape into two equal parts intersect each other. In physics, the term centroid is interchangeable with the center of mass for objects of uniform density, which is the point within an object where all external forces are said to act. For uniform, twodimensional shapes, the centroid can be determined by a technique known as the plumb line method.
Things You'll Need
 Push pin
 Paper cutout of the shape

Determine if the shape is symmetrical. If the shape has a point, line or plane of symmetry, the centroid is located at the point or in the center of the line or plane. For example, the centroid of a uniform sphere is at the center of the sphere. This is the same for uniform cubes, cylinders, circles, squares, triangles and other symmetrical shapes.

Insert the pin near the perimeter of the shape, sticking it to a surface vertically so that it can rotate around the pin. Draw a straight line from the pin to the bottom of the shape.

Remove the pin and place it at another point on the perimeter, then trace a straight line from the pin to the bottom of the shape. The centroid, or center of mass, of the shape is the point where the two lines intersect.