The center of mass of the plate (or the centroid of ) is located at the point , where

and .

Theorem of Pappus :

Let be a plane region that lies entirely on one side of a line in the plane. If is rotated about , then the volume of the resulting solid is the product of the area of and the distance traveled by the centroid of .

i.e, .

Find the volume of a sphere of radius .

From the theorem of Pappus, .

There is no need to use the formula to calculate because, by the symmetry principle, the center of mass must lie on the -axis, so .

The -coordinate of the centroid of the region is .

To find draw a semicircle of radius .

Find the area of the region by using area of semicircle formula.

Therefore, area of the secircular plate is .

Equation of semicircle of radius is .

Thus, .

Consider .

Thus, the center of mass is located at the point .

If the plate is rotated around the -axis get a sphere.

The plates centroid traces a circle of radius .

The circumference of that circle is .

From the theorem of Pappus, .

The volume of of a sphere of radius is .