In this lesson:
By the end of this lesson, you will be able to:
You know these shapes from everyday life:
Key question: Does the cutting angle change the shape?
Think about it before we define anything.
A cross-section is the 2D shape revealed when a plane intersects a 3D object
From daily life: tree rings, CT scans, cheese slices
Three cuts, three different shapes — same cylinder
The same terms apply to any 3D solid.
These words describe the plane's orientation, not the solid's orientation.
A log is being cut at a 45-degree angle to the grain — an oblique cut.
What shape does the cut face have?
A) Circle B) Ellipse C) Rectangle
Think about it — then advance for the answer.
B) Ellipse ✓
Watch out: Tilting the plane does NOT keep the circle — it stretches it into an ellipse
Every cross-section of a sphere is a circle — regardless of angle or position
The sphere is unique: no other standard solid gives only one cross-section shape.
We'll systematically work through each solid, varying the cutting plane.
The five solids:
For each: what shapes are possible? What determines which one you get?
A plane cuts a cylinder parallel to its axis — but not through the center.
What shape is the cross-section?
A) Circle B) Ellipse C) Rectangle (narrower than if through center)
Visualize the cut before advancing.
C) Narrower Rectangle ✓
Watch out: Cross-sections don't have to pass through the center — off-center cuts are valid
Depending on the angle, a cone can produce four different curves:
These are the conic sections — you'll study them in Algebra II.
A horizontal plane cuts a cone halfway between the base and the apex.
How does the cross-section's radius compare to the base radius?
A) Same radius as the base B) Half the base radius C) One-quarter the base radius
Think about how a cone's width changes with height — then advance.
B) Half the base radius ✓
All cross-sections of a sphere are circles — any angle, any position.
No other standard solid has this property.
Surprise: Slice a cube through the midpoints of six edges → regular hexagon
Cutting a cube through the midpoints of six edges produces a regular hexagon
Why?
This is one of geometry's elegant surprises.
Similar to the cone: horizontal slices scale down toward the apex.
Match each solid to its possible cross-sections:
Fill in at least two shapes per solid — then advance for the answers.
✓ A cross-section is the 2D shape produced when a plane cuts a 3D object
✓ The same solid can yield many different cross-sections — angle and position both matter
✓ Sphere = always a circle; cone = richest variety including all conic sections
✓ Off-center cuts are valid — the plane can be anywhere
"A cylinder always gives circles" — only horizontal cuts do; tilted cuts give ellipses
"An angled cylinder cut gives a circle" — oblique cuts give ellipses, not circles
"Cross-sections must pass through the center" — any position is valid
"More cuts, same shape" — the same solid gives many shapes; orientation determines shape
Lesson 2: Solids of Revolution
Today you sliced 3D into 2D. Next, you'll build 3D from 2D.
Click to begin the narrated lesson
Identify cross sections