# Theorems of Pappus on Surfaces of Revolution

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Long before the invention of calculus, Pappus of Alexandria (ca. Century 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 .

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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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