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 solid with the vertices , , and .

From the theorem of Pappus, .

Draw a triangle with the vertices , , and .

By the symmetry, .

The -coordinate of the centroid of the region is .

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

.

Area of the triangle is half the base (left side) times height (left to right).

.

Find the equations for top and bottom sides.

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

The -coordinate of the centroid of the region is .

 Therefore, centroid of the triangle is .

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

The circumference of the circle is .

Area of the triangle is

From the theorem of Pappus, .

The volume of the solid is .