Cavalieri's Principle and Cross-Sections
Apply Cavalieri's Principle: solids with equal-area cross-sections at every height have equal volumes. Visualize cross-sections of 3D solids and understand how this principle justifies the volume formulas for oblique solids.
A lesson of Geometry, in Area and Volume, about 10 minutes.
By the end of this lesson, you will be able to state Cavalieri's Principle, apply it to explain volume formulas, identify cross-sections of common solids, and use cross-sections to solve problems.
The Big Idea
Cavalieri's Principle justifies why the volume formula $V = Bh$ works even for oblique (tilted) solids — because what matters is not the shape's position but the areas of its horizontal slices. A tilted stack of coins has the same volume as a straight stack, because each slice (coin) has the same area at the same height.
Cavalieri's Principle
If two solids have the same height, and if every cross-section parallel to the bases at the same height has equal area, then the solids have equal volume.
2D Analog: If two regions in a plane have equal-length cross-sections at every height, they have equal area. This explains why all triangles with the same base and height have the same area, regardless of shape.
Cross-Sections of Common Solids
A cross-section is the intersection of a solid with a plane.
| Solid | Horizontal cross-section |
|---|---|
| Cube | Square |
| Rectangular prism | Rectangle |
| Cylinder | Circle |
| Cone | Circle (decreasing to point) |
| Sphere | Circle (max at equator) |
| Pyramid | Similar polygon (decreasing to point) |
Vertical cross-sections of the same solids give different shapes (rectangles, triangles, ellipses).
Example 1 — Oblique Cylinder
An oblique cylinder (tilted) has radius 4 and height 10. Find its volume.
By Cavalieri's Principle: every horizontal cross-section is a circle of radius 4 (same as a right cylinder with the same base and height).
$$V = \pi(16)(10) = 160\pi$$
Same as the right cylinder! The tilt doesn't change the volume.
Example 2 — Sphere Cross-Sections
A sphere of radius 5 is cut by a plane 3 units from the center. Find the area of the cross-section.
The cross-section is a circle. Its radius: $r = \sqrt{5^2 - 3^2} = \sqrt{16} = 4$.
Area: $\pi(4)^2 = 16\pi$
Example 3 — Cavalieri Applied to Pyramids
Prove using Cavalieri's Principle that a pyramid has $\frac{1}{3}$ the volume of the corresponding prism.
At height $y$ from the base (base has linear dimension $s$, total height $h$):