A Generalized Cavalieri-Zu Principle

Sidney Kung
10 October, 2012

The Cavalieri-Zu Principle states [1, 2]:

  1. If two plane regions are included between a pair of parallel lines and every line parallel to these lines intersects both regions in line segments of equal length, then the two regions have equal areas.

  2. If two solids in three-space are included between two parallel planes and every plane parallel to these two planes intersects both solids in cross-sections of equal area, then the two solids have equal volume.

Falling into the category of "interesting facts we should have explored but never did" is the fact that the above assertions can be extended to more than two plane regions (solids) for which the said lengths (areas) of segments (cross-sections) need not be equal.

Our generalized version of the Cavalieri-Zu Principle is the following:

G1. If \(n\) plane regions \(P_{1},P_{2}, \ldots , S_{n}\) are included between a pair of parallel lines and every line parallel to these lines intersects the \(n\) regions in line segments \(L_{1},L_{2}, \ldots ,L_{n}\) such that

\(\displaystyle L_{n}=\sum^{n-1}_{i=1}r_{i}L_{i},\)

where \(r_{i},\space i=1,2,\ldots ,n-1\) nonzero constants, then

(*) \(\displaystyle A_{n}=\sum^{n-1}_{i=1}r_{i}A_{i},\)

where \(A_{i}\) is the area of \(P_{i},\space i=1,2,\ldots ,n-1\).

G2. If \(n\) solids \(S_{1},S_{2}, \ldots , S_{n}\) are included between a pair of parallel planes and every plane parallel to these planes intersects the \(n\) solids in plane areas \(A_{1},A_{2}, \ldots ,A_{n}\) such that

\(\displaystyle A_{n}=\sum^{n-1}_{i=1}q_{i}A_{i},\)

where \(q_{i},\space i=1,2,\ldots ,n-1\) nonzero constants, then

(**) \(\displaystyle V_{n}=\sum^{n-1}_{i=1}q_{i}V_{i},\)

where \(V_{i}\) is the area of \(S_{i},\space i=1,2,\ldots ,n-1\).

We use examples to show geometry teachers and students how to apply G1 and G2 to find areas and volumes of geometric figures without relying on methods of calculus.

References

  1. Ji-Huan He, Zu-Geng's axim vs Cavalieri's theory, Applied Mathematics and Computations, 152 (2004) 9-15.
  2. Sidney H. Kung, Proof Without Words: The volume of an ellipsoid via Cavalieri's Principle, College Math. J., 39(3) (May 2008) 190.

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