Cavalieri's Principle and Volume
Aligned to G-GMD.A.2 — Common Core State Standards for Mathematics.
What this lesson teaches
Cavalieri's principle says two solids with the same height have equal volume if every horizontal cross-section at the same level has equal area. This justifies volume formulas: a slanted (oblique) prism has the same volume as a straight one, and the sphere volume can be argued by comparing cross-sections.
Worked example
Argue a cylinder's volume is base times height using Cavalieri. Compare a straight cylinder to a slanted one with the same circular base and height. At every height, each has the identical circular cross-section of area pi*r^2. Since the cross-sectional areas match at every level and the heights are equal, Cavalieri says the volumes are equal: both are pi*r^2*h.
Practice questions
- Two stacks of coins have the same height and identical coins but one leans. Why are their volumes equal?
- State what must be true about cross-sections for Cavalieri's principle to apply.
- A sphere of radius r has cross-sections matching a cylinder-minus-cone solid. What volume does this argument give for the sphere?
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Principio de Cavalieri y volumen
El principio de Cavalieri dice que dos sólidos con la misma altura tienen igual volumen si toda sección transversal horizontal al mismo nivel tiene igual área. Esto justifica fórmulas de volumen: un prisma inclinado (oblicuo) tiene el mismo volumen que uno recto, y el volumen de la esfera se puede argumentar comparando secciones transversales.
Ejemplo: Argumenta que el volumen de un cilindro es base por altura usando Cavalieri. Compara un cilindro recto con uno inclinado con la misma base circular y altura. A cada altura, ambos tienen la misma sección transversal circular de área pi*r^2. Como las áreas de sección coinciden en cada nivel y las alturas son iguales, Cavalieri dice que los volúmenes son iguales: ambos son pi*r^2*h.
More Geometric Measurement & Dimension lessons
Teach this lesson today
An AI tutor that guides with questions instead of just giving answers — all K–12 core subjects, every child in the family.
Start free — no card required