What you’ll learn
Understand Cavalieri's principle.
Key idea
Cavalieri requires area equality at every height, not only the base or a few sampled rows.
Equal bases or finite sample slices do not establish all-level section equality; shapes need not be congruent.
Worked example
Classify examples and non-examples for the concept 'Understand Cavalieri's principle'.
A straight prism and an oblique prism lie between the same parallel planes separated by 8 units. At every height their parallel rectangular sections are 5 by 1 and 5/3 by 3, respectively. Does Cavalieri guarantee equal volumes? Give the full evidence.
- The common perpendicular height is 8; each section has area 5*1=(5/3)*3=5.
- The all-level equality, not identical section outlines, satisfies Cavalieri.
Answer: Yes — both heights are 8 and every paired section area is 5, regardless of the slant or section shape.
Quick Check
Classify examples and non-examples for the concept 'Understand Cavalieri's principle'.
A straight prism and an oblique prism lie between the same parallel planes separated by 6 units. At every height their parallel rectangular sections are 8 by 1 and 8/3 by 3, respectively. Does Cavalieri guarantee equal volumes? Give the full evidence.