Pappus's centroid theorem
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Pappus's centroid theorem consists of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution. The theorem is also known as the Guldinus theorem, Pappus-Guldinus theorem or Pappu's theorem.
The theorem is attributed to Pappus of Alexandria and Paul Guldin.
The first theorem
The first theorem states that the surface area A of a surface of revolution generated by rotating a plane curve C about an axis external to C and on the same plane is equal to product of the arc length s of C and the distance d1 traveled by its centroid.
- <math>A = sd_1.\,<math>
For example, the surface area of the torus with minor radius r and major radius R is
- <math>A = (2\pi r)(2\pi R) = 4\pi^2 R r.\,<math>
The second theorem
The second theorem states that the volume V of a solid of revolution generated by rotating a plane figure F about an external axis is equal to the product of the area A of F and the distance d2 traveled by its geometric centroid.
- <math>V = Ad_2.\,<math>