Theorems of Pappus on Surfaces of Revolution

Long before the invention of calculus, Pappus of Alexandria (ca. 290-350 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.
The controls enable you to choose a rectangular, elliptical or triangular cross section of varying dimensions and . Also you can vary the radius and the angle of rotation θ, up to a complete circle of 360º.

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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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