Cogito
Geometry · Chapter 8 · Lesson 3
Cross Sections and Cavalieri
Slicing a solid, and why a leaning stack holds its volume.
11 problems · about 23 minutes · G-GMD.A.1, G-GMD.B.4
What this lesson teaches
The student identifies cross sections and applies Cavalieri’s principle.
- A cross section is the flat shape left by slicing a solid.
- The angle of the cut changes the shape you get.
- Cavalieri: matching cross sections at every level means equal volume, however much it leans.
Warm Up
Straightforward practice. Get the method working first.
4 problemsA sphere sliced any way. How many sides? Use 0 for a curve.
Answer 0
Why 0 — a circle.
A cube sliced parallel to a face. How many sides has the cross section?
Answer 4
Why A square.
A sphere sliced any way. How many sides? Use 0 for a curve.
Answer 0
Why Always a circle.
A cone sliced horizontally. How many sides? Use 0 for a curve.
Answer 0
Why A circle.
Build It Up
The same ideas with more to keep track of.
3 problemsDoes leaning a stack change its volume? 1 for yes, 0 for no.
Answer 0
Why Cavalieri.
Match each solid to the cross section of a horizontal slice.
Answer Cylinder → Circle; Cube → Square; Triangular prism → Triangle
Why A prism slices into its own base shape.
A cylinder sliced at a slant. Circle or ellipse? 1 for circle, 2 for ellipse.
Answer 2
Why The slant stretches it.
Stretch Yourself
Mixed problems. Work out what kind of question it is before you start.
4 problemsCavalieri needs the two solids to share what? 1 for height, 2 for weight.
Answer 1
Why Geometry, not mass.
Two prisms, same base area and height. Same volume? 1 for yes, 0 for no.
Answer 1
Why Base times height.
The Slice: A cylinder sliced horizontally. How many sides has the cross section? Use 0 for a curve.
Answer 0
Why 0 — a circle.
The Lean: A leaning prism and an upright one, same height and cross sections. Same volume? 1 for yes, 0 for no.
Answer 1
Why Yes.