Pappus's centroid theorem

A theorem stating that a plane region rotated about an external coplanar axis sweeps out a volume equal to its area multiplied by the distance traveled by its centroid.

Centroid: The geometric center of a shape or mass distribution, defined by averaging its points or mass. Its path during rotation supplies the distance multiplying the region’s area.
Plane region: A portion of a plane bounded or otherwise specified by curves or lines. The theorem begins with such a region as the shape being rotated.
Torus: A ring-shaped surface, or the solid it bounds, commonly generated by rotating a circle around a coplanar external axis. Rotating a disk around an external axis produces a solid torus.
Pappus of Alexandria: A Greek mathematician of late antiquity known for the Mathematical Collection and results in geometry. The volume theorem is traditionally associated with his geometric work.
Solid of revolution: A three-dimensional solid formed by rotating a plane curve or region around an axis. The theorem calculates the volume of this kind of solid.
Axis of rotation: A fixed line about which a geometric object or coordinate system is rotated. Its position relative to the region determines whether the theorem applies.
Sphere: The three-dimensional surface whose points are all a fixed distance from one center. A semicircle’s area and centroid path recover the sphere’s volume.
Paul Guldin: A Swiss mathematician and astronomer who published centroid theorems for solids and surfaces of revolution. His name appears alongside Pappus’s in the theorem’s common alternative naming.
Second theorem of Pappus: A theorem relating the surface area of a surface of revolution to the generating curve’s length and centroid path. It is the companion result for surface area rather than volume.
Distance: A nonnegative measure of separation between points or geometric objects. The centroid’s travel distance is the theorem’s multiplier for area.

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