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. Jones, Alexander ed.
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The theorem of Pappus can be either one of two related theorems that can help us derive formulas for the volumes and surface areas of solids or surfaces of revolution. They are named after Pappus of Alexandria, who worked on them.
The First Theorem of Pappus Watch this short video on the first theorem, or read on below: The first theorem of Pappus tells us about the surface area of the surface of revolution we get when we rotate a plane curve around an axis which is external to it but on the same plane. So the surface area will be The animation below shows how this theorem applies to three surfaces of revolution: an open cylinder, a cone, and a sphere.
Here the centroids are shown by the dots, and are a distance a shown in red from the axis of rotation. The Second Theorem of Pappus The second theorem of Pappus is very similar to the first; It tells us that the volume of a solid of revolution which is generated by rotating a plane figure about an external axis equals the area of the plane figure call this A times the distance d which is traveled by the geometric centroid of F.
This looks almost the same as the formula for the surface area, above. It is so much alike you might begin to wonder why the surface areas and volumes were not always the same. The reason they both give different results is that the centroid of a plane figure is usually different from the centroid of its boundary curve.
The area of F is also not the same as the arc length. References Need help with a homework or test question? With Chegg Study , you can get step-by-step solutions to your questions from an expert in the field. Your first 30 minutes with a Chegg tutor is free!
Pappus's Centroid Theorem