# Cavalieri's Principle

Jump to navigation
Jump to search

## Theorem

Let two solid figures $S_1$ and $S_2$ have equal height.

Let sections made by planes parallel to their bases and at equal distances from the bases always have equal area.

Then the volumes of $S_1$ and $S_2$ are equal.

### Extension

An extension of Cavalieri's Principle is as follows:

Let two solid figures $S_1$ and $S_2$ have equal height.

Let the areas of the sections made by planes parallel to their bases and at equal distances from the bases always have the same ratio.

Then the volumes of $S_1$ and $S_2$ are in that same ratio.

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Source of Name

This entry was named for Bonaventura Francesco Cavalieri.

## Sources

- 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Cavalieri's principle** - 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.14$: Cavalieri ($\text {1598}$ – $\text {1647}$) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Cavalieri, Bonaventura**