11. Solids

New York Geometry - 2020 Edition

Investigation: Cavalieri's principle and the volume of a cylinder

Lesson

To understand this investigation, you'll need to understand the formulas for finding the area of a circle and the volume of a prism.

You've already seen that, although a cylinder isn't technically a prism, the volume of a cylinder can be found similarly. In this investigation, we will learn how that fact is true by applying some mathematical principles.

Named after Bonaventura Cavalieri, this principle states that if solid figures have equal cross-sectional area (taken from parallel planes) throughout, then the solid figures will have the same volume. Explore the applet below to see an application of this principle.

Explore the applet below by dragging Point $J$`J` along the line.

Since the prism and the cylinder have equal areas at each cross section, as well as equal heights, they have the same volume.

Recall that the area of a right rectangular prism is $V=Bh$`V`=`B``h`, where $V$`V` is the Volume, $B$`B` is the area of the base, and $h$`h` is the height of the figure (distance between the two bases). How does this relate to the volume of a cylinder, $V=\pi r^2$`V`=π`r`2? Use Cavelieri's Principle in your argument.

1. Find a stack of coins, CDs, or other very thin circular objects.

2. Calculate the area of one circular object, using the formula for the area of a circle.

3. Calculate the volume of the whole stack.

4. Now rearrange your stack so that the coins aren't perfectly in line. Has the volume of the stack changed? Explain why or why not, using Cavalieri's Principle.

Provide informal arguments for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.