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 the cone with height and base radius .

From the theorem of Pappus, .

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

The -coordinate of the centroid of the region is .

Draw a right triangle with the vertices , , and .

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

.

The above diagram shows half of a side view cross section of the cone.

The equation of the line that passes through the points and .

 Point-slope form of line equation is .

The equation of the line that passes through the points and : .

.

The -coordinate of the centroid of the region is .

If the centroid is rotated around the -axis, the distance travelled is the circumference of a circle with radius .

.

From the theorem of Pappus, .

The volume of the cone with height and base radius is.