Theorems of Pappus on Surfaces of Revolution

​
generating figure
rectangle
ellipse
triangle
R
3
θ (º)
270
a
1.5
b
2
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
R
from its centroid equals the product of the arc length
C
of the generating figure and the distance traversed by the figure's centroid,
2πR
. Thus the area of revolution is given by
2πRC
.
For an rectangle of dimensions
a×b
,
C=2a+2b
. For an isosceles triangle with sides
a
,
a
and
b
,
C=2a+b
. For an ellipse of semimajor and semiminor axes
a
and
b
, respectively,
C=2aE(e)
where
E
is a complete elliptic integral of the second kind and
e
is the eccentricity of the ellipse,
e=
1-
2
a
2
b
. Ramanujan proposed the approximation
C≈π3(a+b)-
(3a+b)(a+3b)

. For
a=b=r
, the ellipse simplifies to a circle, with
C=2πr
, and the surface of revolution becomes a torus.
Pappus's second theorem gives the volume of the surface of revolution as
2πR
multiplied by the area
A
of the generating figure. For the rectangle, ellipse, and triangle,
A
equals
ab
,
πab
, and
1
2
b
2
a
-
2
b
/4
, 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
a
and
b
. Also you can vary the radius
R
and the angle of rotation θ, up to a complete circle of 360º.

Details

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

Permanent Citation

S. M. Blinder
​
​"Theorems of Pappus on Surfaces of Revolution"​
​http://demonstrations.wolfram.com/TheoremsOfPappusOnSurfacesOfRevolution/​
​Wolfram Demonstrations Project​
​Published: June 29, 2009