Cavalieri's Principle

Pronunciation: /ˌkɑ vɑˈlyɛr iz ˈprɪn sə pəl/ ?

A stack of coins stacked evenly, and a stack of the same number of coins stacked unevenly.
Figure 1: Stack of coins. Image courtesy Anton(rp). This image licensed under the terms of GNU Free Documentation License.
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Manipulative 1 - Cavalieri's Principle. Created with GeoGebra.

Cavalieri's principle states that if the areas of the cross sections of two solids are equal, and the height of the two solids are equal, then the volumes of the two solids are equal.[1] Cavalieri's principle is considered to be a significant step in the development of infinitesimal calculus.

A stack of coins (see figure 1) illustrates Cavalieri's principle. Since there are the same number of coins in each stack, the stacks necessarily have the same volume. It doesn't matter how the coins are rearranged, the volume remains constant.

Cavalieri's principle is named after Bonaventura Francesco Cavalieri (1598 - November 30, 1647), an Italian mathematician. However, the first know mention of this principle is in the third century by Liu Hui (??) in his commentary on The Nine Chapters on the Mathematical Art (????).

In manipulative 1, click on the blue points and drag them to change the figure. Note that if the height remains constant, the volume also remains constant. Each 'slice' of the cylinder moves over when changing the angle, but does not change in size.

More Information

  • O'Connor, J J and Robertson, E F. Bonaventura Francesco Cavalieri. Biographie. University of St Andrews, Scotland. 2009-03-12. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cavalieri.html.
  • O'Connor, J J and Robertson, E F. Liu Hui. Biographie. University of St Andrews, Scotland. 2009-03-12. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Liu_Hui.html.
  • O'Connor, J J and Robertson, E F. Nine Chapters on the Mathematical Art. Historical Topic. University of St Andrews, Scotland. 2009-03-12. http://www-history.mcs.st-and.ac.uk/HistTopics/Nine_chapters.html.

Cite this article as:


Cavalieri's Principle. 2010-01-09. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cavalierisprinciple.html.

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2010-01-09: Added "References" (McAdams, David.)
2008-11-12: Initial version (McAdams, David.)

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