Cross-Sections of Three-Dimensional Figures

Lesson 1 of 1

In this lesson:

• Define what a cross-section is
• Identify cross-sections of prisms and pyramids
• Predict shapes from cut descriptions

What You Will Learn Today

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

1. Define a cross-section as the 2D figure formed when a plane intersects a 3D solid
2. Identify cross-sections of a right rectangular prism for horizontal, vertical, and diagonal cuts
3. Identify cross-sections of a right rectangular pyramid for horizontal and vertical cuts
4. Predict the cross-section shape from a cut description
5. Explain why the cutting plane's angle and position determines the shape

Slices Are Everywhere Around Us

Real-world cross-sections you've seen:

• Bread loaf — each slice exposes a rectangle
• Orange — a horizontal cut reveals a circle
• MRI scan — doctors see flat images of the body's interior
• Blueprints — floor plans are horizontal cross-sections

What shape does each flat cut reveal?

Defining What a Cross-Section Is

A cross-section is the 2D shape formed where a flat plane intersects a 3D solid.

Key Vocabulary for Describing Cuts

• Cutting plane — the flat surface doing the slicing
• Parallel cut — plane runs parallel to a face
• Perpendicular cut — plane is perpendicular to a face
• Diagonal cut — plane is tilted at an angle to the faces

Predicting the First Prism Cross-Section

Consider a right rectangular prism with length , width , and height .

Horizontal cut — the plane is parallel to the base.

Predict: What 2D shape does the cut expose? What are its dimensions?

Check: What Does a Horizontal Cut Expose?

Quick check:

When you slice a right rectangular prism with a horizontal cut, the cross-section is a ________ with dimensions ____ × ____.

(Write your answer before we reveal it.)

Three Ways to Cut a Rectangular Prism

A right rectangular prism can be cut three ways:

Horizontal Cut Exposes the Base Rectangle

• Plane is parallel to the base
• Cross-section is a rectangle with dimensions
• Shape stays the same wherever the horizontal cut is placed

Vertical Cut Produces a Different Rectangle

A vertical cut runs perpendicular to the base — the plane stands up.

• Cut parallel to front face: rectangle with dimensions
• Cut parallel to side face: rectangle with dimensions
• The shape is still a rectangle — but with different dimensions

Diagonal Cut Produces a Parallelogram

• The plane tilts at an angle — not parallel to any face
• It crosses each vertical face at a different height
• The cross-section angles are no longer 90°
• Result: a parallelogram (not a rectangle)

Watch out: only cuts parallel to a face produce rectangles.

Summary Table for All Prism Cuts

Check: Diagonal Cut vs. Rectangle

Think about it:

1. A rectangular prism is cut diagonally. The cross-section is a ________.

2. Why is the diagonal cross-section NOT a rectangle?

(Hint: what angle does the plane make with the vertical faces?)

Practice: Name Each Prism Cross-Section

1. Horizontal cut, square base → _____
2. Vertical cut, parallel to front → _____
3. Diagonal cut, top-left to bottom-right → _____
4. Vertical cut, parallel to side → _____
5. Horizontal cut near the top → _____
6. Diagonal cut, shallow angle → _____

Answers: Check Your Prism Practice

Moving From Prisms to Pyramids

Now we investigate the right rectangular pyramid.

Key difference from prisms:

• A pyramid tapers to a point (apex) above the rectangular base
• Where you cut changes the type of shape, not just the size

Horizontal Cut of a Pyramid

• Plane is parallel to the base
• Cross-section is a rectangle — similar to the base, but smaller
• The closer to the apex, the smaller the rectangle
• At the apex itself: a single point

Vertical Cut Through the Apex

Through the apex:

• Cut passes through the apex and bisects the base
• Cross-section is an isosceles triangle
• Base of triangle = base of pyramid; apex = pyramid's apex

Vertical Cut Not Through the Apex

Off-center vertical cut (does not pass through apex):

• The top edge of the cross-section is shorter than the base edge
• Both top and bottom edges are present (unlike a triangle)
• Cross-section is a trapezoid

The closer to the edge, the thinner the trapezoid becomes.

Position Changes the Pyramid's Cross-Section

Vertical cuts:

• Through apex → triangle
• Off-center → trapezoid

Horizontal cuts:

• Near base → large rectangle
• Near apex → tiny rectangle
• At apex → single point

Check: Identify the Pyramid Cross-Section

Quick check:

1. A horizontal cut halfway up the pyramid → _____
2. A vertical cut exactly through the apex → _____
3. A vertical cut shifted to the left, missing the apex → _____
4. A horizontal cut just below the apex → _____

Practice: Name Each Pyramid Cross-Section

Name the cross-section for each cut:

1. Horizontal cut at 3/4 of the height up the pyramid → _____
2. Vertical cut passing through the apex → _____
3. Vertical cut parallel to a side face, off-center → _____
4. Horizontal cut very near the base → _____

Answers: Check Your Pyramid Practice

Prisms vs. Pyramids: Key Contrast

• Prism: parallel cuts always produce congruent cross-sections
• Pyramid: parallel cuts produce similar but shrinking cross-sections

Mixed Practice: Predict the Cross-Section

Name the cross-section shape:

1. Prism, horizontal cut → _____
2. Pyramid, horizontal cut at midpoint → _____
3. Prism, diagonal cut → _____
4. Pyramid, vertical cut through apex → _____
5. Prism, vertical cut parallel to front → _____
6. Pyramid, vertical cut shifted right → _____

Answers: Check Your Mixed Practice

Summary: Takeaways and Common Mistakes

Prism cross-sections:

• Horizontal or vertical → rectangle
• Diagonal → parallelogram (not a rectangle)

Pyramid cross-sections:

• Horizontal → rectangle (shrinks toward apex)
• Vertical through apex → triangle
• Vertical off-center → trapezoid

Watch out: horizontal cuts of a pyramid give rectangles, not triangles

Next Steps: Surface Area and Volume

Coming up: 7.G.B.6

In the next lesson, you'll:

• Compute surface areas of prisms and pyramids using cross-section shapes
• Find volumes of 3D figures using the areas of cross-sections
• Connect the cross-section rectangle and triangle from today to area formulas