thumb|right|400px|The theorem applied to an open cylinder, cone and a sphere to obtain their surface areas. The centroids are at a distance a (in red) from the axis of rotation.
In mathematics, Pappus's centroid theorem (also known as the Guldinus theorem, Pappus–Guldinus theorem or Pappus's theorem) is either of two related theorems dealing with the surface areas and volumes of surfaces and solids of revolution.
The theorems are attributed to Pappus of Alexandria and Paul Guldin. Pappus's statement of this theorem appears in print for the first time in 1659, but it was known before, by Kepler in 1615 and by Guldin in 1640.
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 the product of the arc length s of C and the distance d traveled by the geometric centroid of C:
<math display="block">A = sd.</math>
For example, the surface area of the torus with minor radius r and major radius R is
<math display="block">A = (2\pi r)(2\pi R) = 4\pi^2 R r.</math>
Proof
A curve given by the positive function <math> f(x) </math> is bounded by two points given by:
<math> a \geq 0 </math> and <math> b \geq a </math>
If <math> dL </math> is an infinitesimal line element tangent to the curve, the length of the curve is given by:
<math display="block"> L = \int_a^b dL = \int_a^b \sqrt{dx^2 + dy^2} = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>
The <math> y </math> component of the centroid of this curve is:
<math display="block"> \bar{y} = \frac{1}{L} \int_a^b y \, dL = \frac{1}{L} \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>
The area of the surface generated by rotating the curve around the x-axis is given by:
<math display="block"> A = 2 \pi \int_a^b y \, dL = 2 \pi \int_a^b y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx </math>
Using the last two equations to eliminate the integral we have:
<math display="block"> A = 2 \pi \bar{y} L </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 d traveled by the geometric centroid of F. (The centroid of F is usually different from the centroid of its boundary curve C.) That is:
<math display="block">V = Ad.</math>
For example, the volume of the torus with minor radius r and major radius R is
<math display="block">V = (\pi r^2)(2\pi R) = 2\pi^2 R r^2.</math>
This special case was derived by Johannes Kepler using infinitesimals.