Cavalieri’s Principle
In short: Cavalieri’s principle states that solids of equal height whose slices have equal area at every level must fill the same volume, however differently they are shaped.
GeometryGeometryG-GMD.A.1
What is Cavalieri's Principle?
Cavalieri’s principle explains why a tidy column of 50 coins keeps its volume when you nudge it sideways into a lean: nothing was added or removed, and every coin-slice still has the same area.
Formally, take two solids standing between one pair of parallel planes. If each plane parallel to that pair slices both solids in regions of equal area, the solids hold equal volume. The idea is why a slanted cylinder has volume \(\pi r^2 h\) with \(h\) the perpendicular height, and why a pyramid with a shifted apex keeps its volume.
How to spot it
- A solid looks pushed over or sheared, such as a leaning stack, an oblique cylinder or a slanted prism.
- A problem gives a slant length and a perpendicular height, and the volume is requested.
- Two different solids are said to have equal heights and matching slice areas at every level, a job for Cavalieri’s principle.
- A volume formula for an upright solid must be justified for a tilted version.
The method
- Pick the matching upright solid with the same base area and the same perpendicular height.
- Check that the slices at every height have equal area in the two solids, as Cavalieri’s principle demands.
- Conclude the volumes match, then use the standard formula for the upright version.
- Use the perpendicular height, never the slanted side, when computing.
Cavalieri's Principle: solved example
Problem
A leaning stack of circular foam discs has discs of radius 5 cm. Its sloping side is 13 cm long, while its perpendicular height from floor to top is 12 cm. Find its volume.
Solution
Every horizontal slice is a disc of area \(\pi(5)^2 = 25\pi\) square centimeters, exactly like the slices of an upright cylinder of the same height. The rule then gives equal volumes.
The slope length 13 cm is irrelevant; only the perpendicular height counts. \(V = 25\pi \times 12 = 300\pi \approx 942.5\). The volume is about 942.5 cubic centimeters.
Try it
Question 1
Two pyramids have equal heights of 9 cm and the same base area of 36 square cm, but one has its apex off to the side. Do they have the same volume?
Show the answer
Yes. Matching heights and matching slice areas at each level give equal volumes, \(\frac{1}{3}(36)(9) = 108\) cubic cm each.
Question 2
A 200-page book is stacked flat, then its pages are shoved so it leans. Does its volume change?
Show the answer
No. Every page keeps its area and thickness, so the total volume stays the same.
Cavalieri's Principle vs. Volume
Volume is the amount of space a solid fills, measured in cubic units. Here is a comparison rule for finding it, not the quantity itself: it tells you that two solids share the same number without computing either one separately. First compare slices, then borrow the formula of the simpler solid.
Keep going
Lessons
Worksheets and quizzes
- Area, Volume and Dissection Arguments – Worksheet
- Area, Volume and Dissection Arguments – Quiz
- Cross Sections and Solids of Revolution – Worksheet
- Cross Sections and Solids of Revolution – Quiz
Related words
Whole dictionary: A–Z · Geometry · Geometry