Mathematics · 7th Grade · Geometry and Construction · Weeks 19-27

Cross Sections of 3D Figures

Students will describe the two-dimensional figures that result from slicing three-dimensional figures.

• Common Core State Standards
• CCSS.Math.Content.7.G.A.3

About This Topic

When a three-dimensional figure is cut by a plane, the resulting two-dimensional shape is called a cross section. This topic requires students to visualize and predict what 2D figure results from slicing prisms, pyramids, cylinders, and cones at different angles , a skill directly addressed in CCSS 7.G.A.3. Students must connect their knowledge of 2D shapes to spatial reasoning about 3D objects, which is often a genuinely new type of thinking for 7th graders.

Cross sections appear in real-world contexts including medical imaging (MRI and CT scans show internal cross sections of the body), architectural blueprints, and everyday food preparation. Grounding this topic in familiar examples helps students build intuition before working with abstract diagrams.

Physical manipulation and peer discussion are especially effective here because spatial prediction is hard to develop through lecture alone. When students make a prediction, then test it with a physical model, they build the mental imagery that transfers to more abstract problems.

Key Questions

1. Predict the shape of a cross-section when a 3D figure is sliced at different angles.
2. Analyze how the orientation of a slice affects the resulting 2D shape.
3. Construct a physical model to demonstrate various cross-sections of a given 3D figure.

Learning Objectives

• Identify the two-dimensional shapes formed by slicing common three-dimensional figures (prisms, pyramids, cylinders, cones) with a plane.
• Analyze how the angle and orientation of a slicing plane affect the shape of the resulting cross-section.
• Predict the cross-section shape for a given slice of a three-dimensional figure.
• Construct physical models to demonstrate and verify predicted cross-sections of three-dimensional figures.

Before You Start

Identifying and Describing 3D Figures

Why: Students need to be able to name and visualize basic properties of prisms, pyramids, cylinders, and cones before they can analyze slices of them.

Identifying and Describing 2D Figures

Why: Students must be able to recognize and name basic two-dimensional shapes (squares, rectangles, triangles, circles, etc.) to identify the resulting cross-sections.

Key Vocabulary

Cross Section	The two-dimensional shape exposed when a three-dimensional object is sliced by a plane.
Plane	A flat, two-dimensional surface that extends infinitely far. In this context, it represents the 'slice' through a 3D object.
Prism	A solid geometric figure whose two end faces are similar, equal, and parallel rectilinear figures, and whose sides are parallelograms.
Pyramid	A polyhedron with a polygonal base and triangular faces that meet at a point (apex).
Cylinder	A solid geometric figure with straight parallel sides and a circular or oval cross section.
Cone	A three-dimensional geometric shape that tapers smoothly from a flat base (usually circular) to a point called the apex or vertex.

Watch Out for These Misconceptions

Common Misconception

Students assume the cross section of any prism or cylinder is always a rectangle, regardless of the angle of the cut.

What to Teach Instead

Show that a diagonal cut through a cylinder produces an ellipse, not a rectangle. Using clay models or dynamic geometry software to make angled cuts helps students see that the shape of the cross section depends on the orientation of the cutting plane, not just the figure itself.

Common Misconception

Students think slicing a pyramid always produces a triangle.

What to Teach Instead

A horizontal cut through a pyramid produces a smaller version of the base polygon (a triangle for a triangular pyramid, a square for a square pyramid), not a triangle. A vertical cut through the apex produces a triangle. Exploring multiple cut orientations physically or with software builds the correct intuition.

Active Learning Ideas

Hands-On: Clay Slicing Lab

Students form clay models of prisms, pyramids, and cones, then use dental floss to make cuts at various angles. Each student sketches their prediction before cutting, then compares the predicted cross section to the actual result. Groups record all findings on a shared class chart organized by figure type and cut angle.

40 min·Small Groups

Think-Pair-Share: Slice Prediction Cards

Display a 3D figure and a dotted line showing the cut plane. Students write their prediction independently, then compare with a partner and discuss any disagreements. The class then sees the actual cross section using a diagram or physical model, and pairs reflect on what they got right or wrong.

20 min·Pairs

Gallery Walk: Cross Section Matching

Post images of 3D figures and their cross sections around the room in a scrambled format. Students circulate with a worksheet, drawing lines to match each figure-and-cut-angle description to the correct 2D cross section shape. Debrief focuses on the cuts that produced unexpected results.

25 min·Pairs

Real-World Connections

• Radiologists use CT scanners to create cross-sectional images of the human body, allowing them to diagnose conditions by visualizing internal structures without surgery.
• Architects and engineers use cross-sections in blueprints to show the internal structure and composition of buildings and bridges, revealing how different components fit together.
• Chefs often cut through food items like cakes or loaves of bread to reveal their internal structure and layers, demonstrating different cross-sectional shapes.

Assessment Ideas

Exit Ticket

Provide students with a diagram of a cube and a line indicating a slice through it. Ask them to draw the resulting cross-section and label its shape. Then, ask them to describe how changing the angle of the slice would change the shape.

Quick Check

Hold up a physical object like an apple or a block of cheese. Ask students to predict the shape of the cross-section if you were to slice it horizontally, vertically, or diagonally. Call on students to share their predictions and reasoning.

Discussion Prompt

Pose the question: 'Can you always get a square cross-section from a cube? What about a circle from a sphere?' Facilitate a class discussion where students use their knowledge of slicing planes and 3D shapes to justify their answers, perhaps using drawings or models.

Frequently Asked Questions

What is a cross section of a 3D figure?

A cross section is the two-dimensional shape you see when a three-dimensional figure is cut by a flat plane. The shape of the cross section depends on both the type of 3D figure and the angle at which it is sliced. For example, slicing a cone horizontally gives a circle, while a diagonal cut gives an ellipse.

What cross section do you get when you cut a rectangular prism?

A straight horizontal or vertical cut parallel to a face of a rectangular prism produces a rectangle. A diagonal cut can produce a parallelogram or, in some cases, a triangle. The key factor is the orientation of the cutting plane relative to the faces of the prism.

Why do 7th graders study cross sections in geometry?

Understanding cross sections builds spatial reasoning, which is foundational for high school geometry, engineering, architecture, and fields like medicine that use imaging technology. CCSS 7.G.A.3 includes cross sections because it develops the ability to mentally connect 2D and 3D representations of the same object.

How does hands-on work with physical models help students learn about cross sections?

Many students find it difficult to visualize what a cut through a 3D object looks like from a diagram alone. Making predictions with clay models and then physically slicing them turns an abstract spatial task into a concrete one. The act of comparing a prediction to an actual result builds the mental imagery needed for more abstract problems.