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Cavalieri's Principle Demonstration

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About This MicroSim

Cavalieri's Principle says that if two solids have the same height and equal cross-sectional areas at every level, they have equal volumes.

The left panel shows a right solid. The right panel shows an oblique solid with the same base and the same vertical height. One cutting plane passes through both at the height you choose, and the measurements below confirm that the two slices have the same area.

With Show slices on, each solid is drawn as a stack of thin slices, like a stack of coins or a deck of cards. Pushing the stack sideways changes its shape, but it does not add or remove any slices, so the volume cannot change. That is the whole argument.

How to Use

  1. Drag Plane height to move the cutting plane through both solids at once and compare the two highlighted slices.
  2. Drag Tilt angle from 0° to 45° and check that the slice area and the volume do not change.
  3. Switch Solid between the rectangular prism and the cylinder (a stack of coins).
  4. Turn Show slices off to see the smooth oblique solid.

Iframe Embed Code

<iframe src="main.html"
        height="597"
        width="100%"
        scrolling="no"></iframe>

Lesson Plan

Grade Level

9-12 (High School Geometry)

Duration

10-15 minutes

Prerequisites

Volume of a right prism and a right cylinder, and the idea of a cross-section.

Bloom's Taxonomy Level

Understand (L2)

Learning Objective

Students will be able to explain Cavalieri's Principle by comparing right and oblique solids with identical cross-sections, and justify why their volumes are equal.

Activities

  1. Predict (3 min): With the tilt at 0°, students predict whether tilting the solid to 45° will increase, decrease, or keep the volume.
  2. Test (5 min): Students move the plane to five heights at three tilt angles and record the slice area each time.
  3. Explain (5 min): Using a real stack of coins or index cards, students explain the principle to a partner in their own words.

Assessment

Students are shown an oblique cylinder with radius 4 cm and vertical height 10 cm and asked for its volume, with a justification that names Cavalieri's Principle and states what must be equal at every height.

References

  1. Wikipedia. Cavalieri's principle.
  2. Common Core State Standards Initiative. High School Geometry: Geometric Measurement and Dimension (HSG-GMD.A.1, HSG-GMD.A.2).
  3. Eves, H. (1991). Two Surprising Theorems on Cavalieri Congruence. The College Mathematics Journal, 22(2), 118-124.