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.
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