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 Math Help > Geometry > Solids > Solid of Rotation

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).

## Surface Area and Volume of a Solid of Rotation

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

 4/(π r²) óôõ r 0 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.

### Pappus' Centroid Theorem

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,

S = sd1 = sX1.

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,

V = Ad2 = AX2.

The following table summarizes the surface areas and volumes calculated using Pappus's Centroid Theorem for various solids and surfaces of revolution.

 solid section s X1 S A X2 V 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 torus inner radius=R-r, outer radius=R+r circle 2 π r R 4 π² R r π r² R 2 π² R r²

### Other Figures

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.

### Internet references

Mathworld -- Pappus's Centroid Theorem

Whistler Alley -- Torus gives the volume of a Torus

### Related pages in this website

Platonic and Archimedean Solids

Tetrahedron

Centroid describes what it is (the balancing point) and how to calculate it using integrals.

Solid figures

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.