Kepler Finds theVolume of a Torus.
Cut the torus with planes on the central axis of the torus.
The resulting slices can be paired so that the two
paired slices make a cylinder with radius
equal to that of the generating circle and height equal the length cut
in the arc of the circle generated by the center of the generating
The sum of these slices will equal the volume of
Thus the voume of the torus is equal to the volume
of the cylinder with base equal to the generating circle (disc) and height
equal to the circumference of the circle generated by the center of the
If the generating circle has radius r,
and the center of the generating circle is R
from the axis of revolution, then
theVolume is (pi)r2 * 2(pi)R
= 2R(pi r