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.