# Theorems of Pappus on Surfaces of Revolution

Requires a Wolfram Notebook System

Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.

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 .

[more]
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.

## Permanent Citation