Theorems of Pappus-Guldinus

The theorems of Pappus-Guldinus were formulated by the Greek geometer Pappus of Alexandria during the 4th century A.D. (about 340 A.D.) and were restated by the Swiss mathematician Paul Guldinus (1640). The two theorems of Pappus-Guldinus describe the area of surface of revolution and the volume of body of revolution by the circular path traversed by their centroid during the revolution.

Theorem 1: Surface of Revolution

For the surface of a surface of revolution generated by the rotation of a plane curve about a non-intersecting axis, the surface area A of the surface of revolution is equal to the product of the curve length L of the generating curve and the travelled distance d of the centroid of the generating curve during the generation of the surface by revolution. And the travelled distance of the centroid by revolution can also be expressed in terms of the perpendicular distance y of the centroid away from the rotating axis.

Surface of Revolution

The area A of the surface of revolution can be determined by integration through the revolution of an elemental segment dL. Imply

The area A can be rearranged in the form of the integral of the first moment of an elemental segment , which can also be expressed in terms of the centrod of the generating curve, Imply

Theorem 2: Body of Revolution

For the body of a body of revolution generated by the rotation of a plane region about a non-intersecting axis, the body volume V of the body of revolution is equal to the product of the area A of the generating plane region and the travelled distance d of the centroid of the generating region during the generation of the body by revolution. And the travelled distance of the centroid by revolution can also be expressed in terms of the perpendicular distance y of the centroid away from the rotating axis.

Body of Revolution

The volume V of the body of revolution can be determined by integration through the revolution of an elemental area dA. Imply

The volume V can be rearranged in the form of the integral of the first moment of an elemental area, which can also be expressed in terms of the centrod of the generating area, Imply

Applications of Theorems of Pappus-Guldinus

The Theorems of Pappus-Guldinus provides a simple relationship between the area of surface of revolution or the volume of body of revolution and the centroid of the generating plane curve or the centroid of the generating plane area. Therefore the Theorems of Pappus-Guldinus can be used to determine the area of surface of revolution and the volume of body of revolution from the generating curve and the generating area accordingly. And the centroid of a generating plane curve and the centroid of a generating plane area can also be determined from the surface of revolution and body of revolution accordingly.