Pappus's centroid theorem

In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.Volume( n {displaystyle n} -solid of revolution of species p {displaystyle p} )Surface area( n {displaystyle n} -solid of revolution of species p {displaystyle p} )They who look at these things are hardly exalted, as were the ancients and all who wrote the finer things. When I see everyone occupied with the rudiments of mathematics and of the material for inquiries that nature sets before us, I am ashamed; I for one have proved things that are much more valuable and offer much application. In order not to end my discourse declaiming this with empty hands, I will give this for the benefit of the readers:The ratio of solids of complete revolution is compounded of (that) of the revolved figures and (that) of the straight lines similarly drawn to the axes from the centers of gravity in them; that of (solids of) incomplete (revolution) from (that) of the revolved figures and (that) of the arcs that the centers of gravity in them describe, where the (ratio) of these arcs is, of course, (compounded) of (that) of the (lines) drawn and (that) of the angles of revolution that their extremities contain, if these (lines) are also at (right angles) to the axes. These propositions, which are practically a single one, contain many theorems of all kinds, for curves and surfaces and solids, all at once and by one proof, things not yet and things already demonstrated, such as those in the twelfth book of the First Elements. In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorems are attributed to Pappus of Alexandria and Paul Guldin. The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to the product of the arc length s of C and the distance d traveled by the geometric centroid of C: For example, the surface area of the torus with minor radius r and major radius R is The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d traveled by the geometric centroid of F. (Note that the centroid of F is usually different from the centroid of its boundary curve C.) That is: For example, the volume of the torus with minor radius r and major radius R is This special case was derived by Johannes Kepler using infinitesimals. Let A {displaystyle A} be the area of F {displaystyle F} , W {displaystyle W} the solid of revolution of F {displaystyle F} , and V {displaystyle V} the volume of W {displaystyle W} . Suppose F {displaystyle F} starts in the x z {displaystyle xz} -plane and rotates around the z {displaystyle z} -axis. The distance of the centroid of F {displaystyle F} from the z {displaystyle z} -axis is its x {displaystyle x} -coordinate