Curriculum placement
Geometry. This describes the skill, not a learner estimate.
Cavalieri's principle
Use equal cross-sectional areas to justify solid volumes including spheres
Problem types
- Equal cross-sections, equal volumes
- Argue for the sphere's volume with Cavalieri's principle
- Check the slices: equal areas at a chosen height
- Cavalieri arguments for oblique solids, cones and pyramids
Builds on
- Why the circle and solid formulas hold
- Cross-sections of solids · Earlier idea
Slicing a solid gives cross-sections at every height, and equal cross-sections at every height give equal volumes.
Leads to
- Modeling density · See it applied
Solids with equal cross-sections have equal volumes, so a slanted and a straight block of the same material have equal mass.
- Volumes of cylinders, cones, pyramids, spheres · See it applied
Equal cross-sections show that an oblique solid has the volume of the right solid with the same base and perpendicular height.
- Volume of cylinders, cones, and spheres · See it applied
Equal cross-sections explain why the volume of a cone and a sphere follow from simple comparisons.
Related skills
- Cross-sections and solids of revolution · Related idea
A solid of revolution is built from circular cross-sections, and Cavalieri compares solids by their cross-sections.
- Volume in unit cubes · Related idea
Stacking layers of unit cubes is the idea behind Cavalieri: solids with equal layers at each height have equal volume.
Open the interactive map to explore lessons and skill connections.