Cross-Sections of 3D Objects

Lesson 1 of 2: Slicing Three-Dimensional Shapes

In this lesson:

• Identify the 2D shape produced when a plane slices a 3D object
• Predict how cross-section shape changes with cutting angle

Learning Objectives

By the end of this lesson, you will be able to:

1. Identify the 2D cross-section shape produced when a plane slices a standard 3D object
2. Predict how the cross-section shape changes depending on the cutting plane's angle and position

Same Object, Different Slice

You know these shapes from everyday life:

• Slicing a loaf of bread → rectangular face
• Slicing an orange in half → circular face
• Cutting a log at an angle → ?

Key question: Does the cutting angle change the shape?

Think about it before we define anything.

What Is a Cross-Section?

A cross-section is the 2D shape revealed when a plane intersects a 3D object

• Think of it as the "face" you see at the cut
• The shape depends on: (1) the object, and (2) the plane's orientation
• Same object + different angle = different cross-section

From daily life: tree rings, CT scans, cheese slices

One Cylinder, Three Different Cross-Sections

Three cuts, three different shapes — same cylinder

Vocabulary: Three Types of Cuts

• Horizontal cross-section — plane parallel to the base
• Vertical cross-section — plane perpendicular to the base
• Oblique cross-section — plane at an angle (neither horizontal nor vertical)

The same terms apply to any 3D solid.

These words describe the plane's orientation, not the solid's orientation.

Quick Check

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.

Answer: Ellipse

B) Ellipse ✓

• Horizontal cut (perpendicular to axis) → circle
• Oblique cut (tilted) → ellipse
• Why? The tilt stretches one dimension of the circle while leaving the other unchanged

Watch out: Tilting the plane does NOT keep the circle — it stretches it into an ellipse

The Sphere: A Special Case

Every cross-section of a sphere is a circle — regardless of angle or position

• Cut through the center → great circle (maximum radius = )
• Cut off-center at distance → smaller circle with radius

The sphere is unique: no other standard solid gives only one cross-section shape.

Now: Five Standard Solids

We'll systematically work through each solid, varying the cutting plane.

The five solids:

1. Cylinder
2. Cone
3. Sphere (already covered)
4. Rectangular prism (box)
5. Square pyramid

For each: what shapes are possible? What determines which one you get?

Cylinder: All Cross-Sections

• Perpendicular to axis → circle (radius = )
• Parallel to axis → rectangle (width = , height = )

Quick Check: Off-Center Cylinder Cut

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.

Answer: Narrower Rectangle

C) Narrower Rectangle ✓

• The cutting plane is parallel to the axis → always a rectangle
• Not through the center → the width = length of the chord at that offset
• Same height as the cylinder; narrower width

Watch out: Cross-sections don't have to pass through the center — off-center cuts are valid

Cone: Cross-Sections

• Horizontal (perpendicular to axis) → circle (smaller higher up)
• Vertical through apex → isosceles triangle
• Oblique → ellipse

Conic Sections: A Preview

Depending on the angle, a cone can produce four different curves:

These are the conic sections — you'll study them in Algebra II.

Quick Check: Cone at Half-Height

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.

Answer: Half the Base Radius

B) Half the base radius ✓

• The cone tapers linearly: radius scales proportionally with distance from the apex
• At height from the apex: radius =
• At height from the apex: radius = (the base)

Sphere Cross-Sections (Revisited)

All cross-sections of a sphere are circles — any angle, any position.

• Through center → great circle with radius
• Distance from center → smaller circle, radius

No other standard solid has this property.

Rectangular Prism: Surprising Variety

• Parallel to a face → rectangle (same dimensions as that face)
• Diagonal through four edges → parallelogram or rectangle
• Through exactly three faces → triangle

Surprise: Slice a cube through the midpoints of six edges → regular hexagon

Cube: The Hexagonal Cross-Section

Cutting a cube through the midpoints of six edges produces a regular hexagon

Why?

• The plane hits all 6 faces at the same distance from each vertex
• By symmetry, all six cut edges are equal length
• Equal sides + all interior angles equal → regular hexagon

This is one of geometry's elegant surprises.

Square Pyramid: Cross-Sections

• Horizontal (parallel to base) → smaller square (closer to apex = smaller)
• Vertical through apex → triangle (isosceles)
• Vertical parallel to a face → trapezoid

Similar to the cone: horizontal slices scale down toward the apex.

Cross-Section Classification Challenge

Match each solid to its possible cross-sections:

Fill in at least two shapes per solid — then advance for the answers.

Cross-Section Summary Table

Key Takeaways

✓ 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

Watch Out: Common Mistakes

"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

What's Next

Lesson 2: Solids of Revolution

• We reversed the question: given a 2D shape, spin it to build a 3D solid
• A rectangle spins into a cylinder; a right triangle spins into a cone
• Connect cross-sections and solids of revolution

Today you sliced 3D into 2D. Next, you'll build 3D from 2D.