Key takeaway: Cavalieri’s principle states that if two solids stand equally tall and every horizontal slice of one matches the area of the matching slice of the other, their volumes are equal.
Grade 10 – GeometryG-GMD.A.1Geometry and Measurement
Definition of Cavalieri's Principle
Cavalieri’s principle is easiest to see with a stack of coins. Push the stack sideways so it leans. The pile looks different, yet it holds the same coins, so it takes up the same space.
Formally, cut two solids with planes parallel to their bases. Whenever the slices at one height have equal areas, the two volumes agree. That explains why a leaning cylinder uses the upright volume formula with its straight-up height, and why shoving the top of a prism sideways costs no volume.
Formula and notation
In symbols, Cavalieri’s principle for solids \(S_1\) and \(S_2\) of equal height reads: if the slice areas at height \(t\) agree,
\[ A_1(t) = A_2(t) \ \text{for every } t \ \Longrightarrow \ V_1 = V_2 \]
For any cylinder with base area \(B\) and perpendicular height \(h\), the volume is \(V = B\,h\).
Key facts
- Heights are taken straight up from the base, never along a leaning side.
- Sliding each layer sideways reshapes the stack, yet the space it fills stays put.
- The idea justifies the volume formulas for oblique prisms and cylinders.
- It was used to show that a pyramid has one third of the volume of a prism.
Cavalieri's Principle example
A leaning cylinder-shaped candle has a circular base of radius 4 cm. Its sloped side is 17 cm long, but its top is exactly 15 cm above the base plane. Find its volume.
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Cross-sections parallel to the base are circles of radius 4 cm at every height, just like a straight cylinder of height 15 cm.
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Base area: \(B = \pi(4)^2 = 16\pi\) cm².
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Volume with the perpendicular height: \(V = 16\pi \times 15 = 240\pi \approx 754\) cm³.
The candle holds \(240\pi \approx 754\) cm³ of wax.
Common mistake
Watch out
Pilar multiplies the base area by the 17 cm slanted side and gets \(16\pi \times 17 = 272\pi\) cm³.
Why it is wrong
Cavalieri’s principle uses height measured at a right angle to the base, 15 cm. The 17 cm slant is longer than the true height, so it overstates the volume. The volume stays \(240\pi\) cm³, the same as for an upright candle of height 15 cm.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement