
Re: Pappus Theorem
Posted:
May 1, 1998 7:10 AM


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 2space. > 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 semicircular 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 semicircular 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.
Peter

