Pappus's Theorems

Theorem

There are $2$ theorems which bear the name of Pappus of Alexandria:


Pappus Centroid Theorems, also known as the Pappus-Guldinus Theorems

These comprise the $2$ separate theorems:


First Pappus-Guldinus Theorem

Let $C$ be a plane figure that lies entirely on one side of a straight line $\LL$.

Let $S$ be the solid of revolution generated by $C$ around $\LL$.


Then the volume of $S$ is equal to the area of $C$ multiplied by the distance travelled by the centroid of $C$ around $\LL$ when generating $S$.


Second Pappus-Guldinus Theorem

Let $C$ be a plane figure that lies entirely on one side of a straight line $L$.

Let $S$ be the solid of revolution generated by $C$ around $L$.

Then the surface area of $S$ is equal to the perimeter length of $C$ multiplied by the distance travelled by the centroid of $C$ around $L$ when generating $S$.


Pappus's Hexagon Theorem

Let $A, B, C$ be a set of collinear points.

Let $a, b, c$ be another set of collinear points.

Let $X, Y, Z$ be the points of intersection of each of the straight lines $Ab$ and $aB$, $Ac$ and $aC$, and $Bc$ and $bC$.


Then $X, Y, Z$ are collinear points.


Also known as

can also be seen rendered as Pappus' theorems.


Source of Name

This entry was named for Pappus of Alexandria.


Sources

  • 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Pappus' theorems
  • 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Pappus' theorems
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