Informal volume arguments (Cavalieri's principle, dissection)
Cavalieri's Principle
Cavalieri's Principle is a topic in Three-Dimensional Measurement in the Common Core State Standards. It is aligned to Standard G-GMD.1, G-GMD.2, which requires students to give an informal argument for the volume formulas of solids using dissection and Cavalieri's principle.
Cavalieri's principle states that solids with equal cross-sectional areas at every height have equal volume.
Theory
Volume formulas can be justified with informal arguments:
- Cavalieri's principle: if two solids have the same cross-sectional area at every height, they have the same volume — even if one is slanted (oblique).
- Dissection: cut a solid into pieces and rearrange them into a familiar solid of equal volume.
These explain why an oblique prism has the same volume as an upright one (\(V=Bh\)) and why the sphere and cone formulas hold.
Cavalieri's principle:
How to apply the principle
- Compare cross-sections at each height of two solids.
- If they match everywhere, the volumes are equal.
- Use \(V=Bh\) for oblique prisms and cylinders with the perpendicular height.
If two solids have equal cross-sectional areas at every height, they have the same volume.
By Cavalieri, an oblique prism has the same volume as an upright one: \(V=Bh\).
| \(V\) | \(=\) | \(12\times 5=60\) |
No. By Cavalieri's principle, the volume depends only on the base area and the (perpendicular) height, not the slant.
Cutting a solid into pieces and rearranging them into a familiar solid of the same volume.
Common pitfalls
Frequently asked questions
What is Cavalieri's principle?
If two solids have equal cross-sectional areas at every height, they have the same volume.
Does an oblique prism have the same volume as an upright one?
Yes, by Cavalieri's principle: \(V=Bh\) with the perpendicular height.
What is a dissection argument?
Cutting a solid into pieces and rearranging them into a familiar solid of the same volume.
Which height do you use for an oblique solid?
The perpendicular (vertical) height between the bases, not the slanted edge.