Cavalieri's Principle

Pronunciation: /ˌkɑ.vɑˈljɛr.iz ˈprɪn.sə.pəl/ Explain

A stack of coins stacked evenly, and a stack of the same number of coins stacked unevenly.
Figure 1: Stack of coins illustrating Cavalieri's principle.
Click on the blue points and drag them to change the figure.

Why are the volumes of the two cylinders always the same?
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 heights of the two solids are equal, then the volumes of the two solids are equal. 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 (劉徽) in his commentary on The Nine Chapters on the Mathematical Art (九章算術).

References

  1. McAdams, David E.. All Math Words Dictionary, Cavalieri's principle. 2nd Classroom edition 20150108-4799968. pg 30. Life is a Story Problem LLC. January 8, 2015. Buy the book

More Information

  • O'Connor, J J and Robertson, E F. Bonaventura Francesco Cavalieri. Biographie. University of St Andrews, Scotland. 6/27/2018. http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Cavalieri.html.

Cite this article as:

 McAdams, David E. Cavalieri's Principle. 3/1/2019. All Math Words Encyclopedia. Life is a Story Problem LLC. http://www.allmathwords.org/en/c/cavalierisprinciple.html.

Image Credits

  • All images and manipulatives are by David McAdams unless otherwise stated. All images by David McAdams are Copyright © Life is a Story Problem LLC and are licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
  • Stacks of coins: Anton, http://commons.wikimedia.org. Licensed under the terms of the GNU Free Documentation License, Version 1.2 or any later version.

Revision History

3/1/2019: Restored Chinese characters. (McAdams, David E.)
12/21/2018: Reviewed and corrected IPA pronunication. (McAdams, David E.)
6/25/2018: Removed broken links, updated license, implemented new markup, updated GeoGebra apps. (McAdams, David E.)
1/9/2010: Added "References". (McAdams, David E.)
11/12/2008: Initial version. (McAdams, David E.)

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