Cavalieri’s Principle
Solids with equal cross-sectional areas at every height have equal volume.
Key idea
Solids with equal cross-sectional areas at every height have equal volume.
In depth
If two solids have equal height and equal cross-sectional areas at every level, they have equal volume. A skewed stack of coins has the same volume as a straight one.
Worked example
An oblique cylinder and a right cylinder with equal radius and height have the same volume πr²h.
Real-world uses
- Explaining why a pyramid and a prism with the same base and height have related volumes
Common mistake
Using the slant length instead of the perpendicular height.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Cavalieri’s Principle?
Solids with equal cross-sectional areas at every height have equal volume. It is a topic at the expert level in Geometry.
Where is Cavalieri’s Principle used in real life?
Real-world examples: Explaining why a pyramid and a prism with the same base and height have related volumes.
What should I learn before Cavalieri’s Principle?
Make sure you are comfortable with Cross Sections, the previous topic in Geometry.
What is a common mistake with Cavalieri’s Principle?
Using the slant length instead of the perpendicular height.
How can I practise Cavalieri’s Principle?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Cavalieri’s Principle?
Move on to Translations next.
Related expert topics in Geometry
- Power of a Point – For a point and a circle, products of segment lengths along secants are equal.
- Cyclic Quadrilaterals – Opposite angles of a quadrilateral inscribed in a circle sum to 180°.
- Area by Coordinates (Shoelace) – Area of a polygon from vertices.
- Parabola (Geometric Definition) – Points equidistant from a focus and a directrix.
- Ellipse – Sum of distances to two foci is constant.
- Hyperbola – Difference of distances to two foci is constant.