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Pappus' Theorems


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The following two theorems are about surfaces and solids of revolutions:

Theorem

Suppose that an arc of a plane curve is rotated through one revolution about a line in the plane that does not cut the arc. Then the area of the surface of revolution obtained is equal to the length of the arc times the distance travelled by the centroid of the curve.

Theorem

Suppose that a plane region is rotated through one revolution about a line in the plane that does not cut the region. Then the volume of the solid of revolution obtained is equal to the area of the region times the distance travelled by the centroid of the region.

The theorems can be used to find surface areas and volumes, such as those of a torus. They can also be used to find the positions of centroids. For example, using the second theorem and the known volume of a sphere, the position of the centroid of the region bounded by a semicircle and a diameter can be found.

Subjects: Mathematics.


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