Pappus's area theorem

For a plane curve rotated about an external coplanar axis, the generated surface area equals the curve’s length multiplied by the distance traveled by its centroid.

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Centroid: The geometric center of a figure or the average position of its points, weighted by length, area, or mass. The curve’s centroid determines the radius of its circular path during rotation.
First moment of a curve: The integral of position along a curve, used to locate its length-weighted centroid. It provides the calculation that locates the centroid of the generating curve.
Torus: A ring-shaped surface or solid formed by revolving a circle around a coplanar axis outside it. Its surface area follows by multiplying the circle’s circumference by its center’s travel distance.
Pappus of Alexandria: A Greek mathematician of late antiquity known for the mathematical collection called the Synagoge. The area theorem bears his name, though its historical attribution is not straightforward.
Surface of revolution: A surface formed by rotating a curve around an axis in its plane. The theorem computes the area of this surface from the generating curve.
Line integral: An integral that accumulates a scalar or vector quantity along a curve. Curve length and its first moment are both expressed through integration along the curve.
Sphere: The set of points in three-dimensional space at a fixed distance from a center. A semicircle rotated about its diameter generates a sphere whose area the theorem recovers.
Paul Guldin: A Swiss mathematician and astronomer who published centroid theorems for solids of revolution in the seventeenth century. His writings helped transmit the centroid theorem now associated with Pappus.
Arc length: The distance measured along a curve between two points. The curve’s arc length supplies the theorem’s other multiplicative factor.
Center of mass: The weighted average position of the mass in a physical system. For a uniform wire shaped like the curve, its center of mass is the curve’s centroid.