# Cavalieri's Principle/Extension

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## Theorem

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

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## Source of Name

This entry was named for Bonaventura Francesco Cavalieri.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.14$: Cavalieri ($\text {1598}$ – $\text {1647}$) - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next):**Cavalieri's principle** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next):**Cavalieri's principle**