Cavalieri's Principle

College/University

Definition

This principle tells us that if there are two solids with the same altitude, the section that come about from the planes that are parallel and the same distance from the two solids' bases are always equal. The volumes of the solids are also the same.

Worked examples

\(\)Cylinder (radius \( r\), height \( h\)) and oblique prism (same base area \( \pi r^2\), height \( h\))\(\)

Every cross-section at the same height has area \(\pi r^2\), so both solids have volume \(\pi r^2 h\).

\(\)Right cone and oblique cone with identical base and height have equal volumes \( \frac{1}{3}\pi r^2 h\)

Even though the oblique cone leans, matching cross-sectional areas guarantee the same volume.

Common mistakes

The two solids must have the same shape to use Cavalieri's Principle. → They need the same height and equal cross-sectional areas at every level — shapes can differ. A cylinder and a prism with matching base area and height satisfy the principle even though their shapes differ.
Cavalieri's Principle only applies to prisms and cylinders. → It applies to any solids with equal cross-sections at matching heights, including cones, pyramids, and spheres. The principle is general — it works for oblique and curved solids, not just right prisms.
Cross-sections must be at the base to apply the principle. → Cross-sections at every height between the bases must be equal in area. The principle requires matching areas at all parallel slices, not just one slice.

Where you'll use it next

You'll use Cavalieri's Principle to derive volume formulas for pyramids, cones, and spheres in geometry, and to justify integration techniques for volumes of revolution in calculus.

Found in 1 StudyPug lesson

Mastering Volumes of Solids with Known Cross-Sections

Calculus 2

Dive into the world of 3D geometry and calculus. Learn how to analyze cross-sections, apply integration techniques, and solve real-world volume problems with confidence and precision.

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