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Topic: Re:Proof of Pappus' Theorem
Replies: 7   Last Post: May 15, 1998 7:13 PM

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Peter Ash

Posts: 13
Registered: 12/6/04
Re: Pappus Theorem
Posted: May 1, 1998 7:10 AM
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NATHANIEL SILVER wrote (in part):

> One needs to find the centroid, which is the center
> of mass or the average x coordinate and the average
> y- coordinate of the region, (x bar, y bar) in 2-space.
> Calculus may come into play here, but the centroid
> usually is the center when that label is used, like for
> ellipses, rectangles, squares, equilateral triangles,
> etc. Then one proceeds as follows.
> Find the distance D between centroid and axis (line)
> of revolution, using a formula for the distance between
> a point and a line. Pappus' formula for volume is
> V = 2pi*D*A where A is the area of the region.

One can also use Pappus' theorem in the reverse manner,
to *find* the centroid of certain regions. For example,
suppose you want to find the centroid of a semi-circular
region (where one side of the region is the diameter AOB).
By symmetry, the centroid must lie somewhere on OC, the
perpendicular bisector of AB. When the semi-circular region
is rotated to form a solid of revolution, the result is a
solid sphere with volume given by v = (4/3)*PI*r^3., and the
area of the region is a = (1/2)*PI*r^2. Plug these values
into Pappus' formula v = 2*PI*d*a and solve for d, which
locates the position of the centroid on OC.


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