A solid of rotation is the figure that results from rotating a plane figure about an external axis (an axis on the same plane as the figure such that no two points of the figure are on opposite sides of the axis).
If you know the centroid of a plane figure, you can use Pappus's Theorem to find the volume of a solid of rotation of that plane figure. (Conversely, if you know the volume of a solid of rotation, you can reverse-engineer the centroid using Pappus's Theorem.)
The x coordinate of the centroid is denoted X, and the y coordinate is denoted y. The centroid X of a finite number of point masses is the sum of the product of each mass and its x-coordinate divided by the sum of all the masses. The centroid of a plane figure is the integral of the x-values of all the slices of the area divided by the total area. For example, the area of a quarter circle of radius r is (1/4) π r², so the centroid of a quarter circle given by y=sqrt(r²-x²) is
x sqrt(r²-x²) dx = 4/(π r²)(1/3) π r³ = (4/3)(r/π)
The centroid of a semicircle is the same, and so this is the value used in the table, below, to calculate the volume of a sphere.
The surface area S of a surface of revolution generated by the revolution of a curve about an external axis is equal to the product of the arc length s of the generating curve and the distance d1 traveled by the curve's centroid X1,
Similarly, the volume V of a solid of revolution generated by the revolution of a lamina about an external axis is equal to the product of the area A of the lamina and the distance d2 traveled by the lamina's centroid X2,
S = sd1 = 2πsX1.
V = Ad2 = 2πAX2.
The following table summarizes the surface areas and volumes calculated using Pappus's Centroid Theorem for various solids and surfaces of revolution.
|cone||right triangle||sqrt(r²+h²)||(1/2) r||π r sqrt(r²+h²)||(1/2)hr||(1/3) r||(1/3)π r²h|
|cylinder||rectangle||h||r||2 p r h||hr||(1/2) r||π r²h|
|sphere||semicircle||π r||2r/π||4 π r²||(1/2)π r²||(4/3)(r/π)||(4/3)π r³|
|circle||2 π r||R||4 π² R r||π r²||R||2 π² R r²|
A student asked me to help him calculate the volume of a solid of rotation in which a semicircle with the "rounded side in" is rotated about an external axis. This is a "meticulously half-eaten torus" which you could also think of as the intersection of a torus with a cylinder.
We solved it using the "shell" method, in which we integrated the area of cylindrical shells of the figure, each with radius x and height 2 sqrt(r^2-(R-x)^2), from R-r to R. We got the right answer, which is pi^2Rr^2-4/3 pi r^3. I was surprised to see the volume of the little sphere as part of the answer, until I reread this page, and I can see why... The other half of that torus would have area pi^2Rr^2+4/3 pi r^3, because it's a semicircle with centroid R+(4/3)r/pi and area (1/2) pi r^2. The result follows immediately from Pappus' theorem.
Mathworld -- Pappus's Centroid Theorem
Whistler Alley -- Torus gives the volume of a Torus
Platonic and Archimedean Solids
Centroid describes what it is (the balancing point) and how to calculate it using integrals.
Pappus-related things you might have been looking for when you found this page.
Pappus Theorem, which is more about points and lines, and is a special case of Pascal's Theorem, in which a hexagon is inscribed in a conic. I link to it here, because you may have been looking for it when you found this page.
Pappus' Chain of Circles -- circles inscribed in an arbelos
The webmaster and author of this Math Help site is Graeme McRae.