Cutting a solid with a plane leaves a flat shape called a cross section.
Step 1: Let's Learn
Read it, or press Listen and follow the words.
The angle matters
A cylinder sliced horizontally gives a circle; sliced at a slant it gives an ellipse.
Cavalieri’s principle
Two solids of the same height whose cross sections match at every level have the same volume.
What that means
A leaning stack of coins holds the same volume as a straight one. Sliding the slices sideways changes nothing.
Slicing a solid
A cross section is the shape revealed by slicing a solid with a plane. A cylinder sliced horizontally gives a circle; sliced vertically, a rectangle; sliced at an angle, an ellipse. The slice direction determines the shape entirely.
Cavalieri's principle
Two solids with equal heights whose cross sections at every level have equal areas have equal volumes. A leaning stack of coins holds the same volume as a straight one, because every horizontal slice is unchanged.
Why the principle matters
It lets you compute the volume of an oblique prism using the ordinary formula, and it is the route to the sphere's volume without calculus. It is a genuine theorem that replaces a great deal of hard work.
Cross sections in practice
Medical scans, engineering drawings and geological surveys all present solids as sequences of cross sections. Reconstructing a three-dimensional object from its slices is a skill with immediate real application.
Step 2: Try It Yourself
Tap and try it out.
Cylinder
Flat faces
2
Does it roll?
Yes
A cylinder has 2 flat faces, like a can.
Step 3: Watch an Example
One step at a time.
Watch Amara Compare Two Stacks
Two stacks of identical coins, one straight and one pushed into a lean.
- Step 1
Both stacks have the same height.
Step 4: Your Turn
Practice makes it stick.
The Slice
Problem 1 of 2
A cylinder sliced horizontally. How many sides has the cross section? Use 0 for a curve.
The Lean
Problem 2 of 2
A leaning prism and an upright one, same height and cross sections. Same volume? 1 for yes, 0 for no.
Slice It
1 of 8
A cube sliced parallel to a face. How many sides has the cross section?
2 of 8
A sphere sliced any way. How many sides? Use 0 for a curve.
3 of 8
A cone sliced horizontally. How many sides? Use 0 for a curve.
4 of 8
Does leaning a stack change its volume? 1 for yes, 0 for no.
5 of 8
Match each solid to the cross section of a horizontal slice.
Tap a card on the left to start.
6 of 8
A cylinder sliced at a slant. Circle or ellipse? 1 for circle, 2 for ellipse.
7 of 8
Cavalieri needs the two solids to share what? 1 for height, 2 for weight.
8 of 8
Two prisms, same base area and height. Same volume? 1 for yes, 0 for no.
Step 5: Quick Check
Show what you know.
Question 1 of 1
A sphere sliced any way. How many sides? Use 0 for a curve.
What You Learned
- 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.