# Cavalieri's Principle

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## 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

## 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 ($\text {1598}$ – $\text {1647}$) - 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Cavalieri, Bonaventura**