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Theorems of Pappus on Surfaces of Revolution

Long before the invention of calculus, Pappus of Alexandria (ca. Century AD) proposed two theorems for determining the area and volume of surfaces of revolution. Pappus's first theorem states that the area of a surface generated by rotating a figure about an external axis a distance from its centroid equals the product of the arc length of the generating figure and the distance traversed by the figure's centroid, . Thus the area of revolution is given by .
For an rectangle of dimensions , . For an isosceles triangle with sides , and , . For an ellipse of semimajor and semiminor axes and , respectively, where is a complete elliptic integral of the second kind and is the eccentricity of the ellipse, . Ramanujan proposed the approximation . For , the ellipse simplifies to a circle, with , and the surface of revolution becomes a torus.
Pappus's second theorem gives the volume of the surface of revolution as multiplied by the area of the generating figure. For the rectangle, ellipse, and triangle, equals , , and , respectively.
Two related results are Pappus's centroid theorems, which involve surfaces generated by rotating about an axis passing through the centroid of the generating figure.

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Here are the theorems of Pappus for the most symmetrical cases, :
Snapshot 1: a square: volume = , area =
Snapshot 2: a torus: generating figure is a circle of radius : volume = , area =
Snapshot 3: an equilateral triangle: volume = , area =
Reference: S. M. Blinder, Guide to Essential Math, Amsterdam: Elsevier, 2008 p. 4.
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