Theorems of Pappus on Surfaces of Revolution

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Long before the invention of calculus, Pappus of Alexandria (ca. 290-350 AD) proposed two theorems for determining the area and volume of surfaces of revolution. Pappus's first theorem states that the area of a surface generated by rotating a figure about an external axis a distance from its centroid equals the product of the arc length
of the generating figure and the distance traversed by the figure's centroid,
. Thus the area of revolution is given by
.
Contributed by: S. M. Blinder (March 2011)
Open content licensed under CC BY-NC-SA
Snapshots
Details
Here are the theorems of Pappus for the most symmetrical cases, :
Snapshot 1: a square: volume = , area =
Snapshot 2: a torus: generating figure is a circle of radius : volume =
, area =
Snapshot 3: an equilateral triangle: volume = , area =
Reference: S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 p. 4.
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