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Back to Describe the two-dimensional figures that result from slicing three-dimensional figures

Cross-Sections of Three-Dimensional Figures

For each problem, read the description of the solid and the cut carefully before naming the cross-section. Drawing a quick sketch may help.

Grade 7·~35 min·Common Core Math - Grade 7·standard·7-g-a-3
Work through problems with immediate feedback
A

Recall / Warm-Up

1.

Which polygon has exactly four sides and four right angles?

2.

Which 3D figure has a rectangular base and triangular faces that all meet at a single point above the base?

3.

Which polygon has exactly two pairs of parallel sides but does NOT necessarily have four right angles?

B

Fluency Practice

Oblique drawing of a rectangular prism with a horizontal cutting plane highlighted at mid-height, connected by an arrow to an isolated cross-section shape.
1.

A right rectangular prism is cut with a horizontal plane parallel to its base. What shape is the cross-section?

Oblique drawing of a rectangular prism with a vertical cutting plane parallel to the front face highlighted, connected by an arrow to an isolated cross-section shape.
2.

A right rectangular prism is cut with a vertical plane perpendicular to the base and parallel to the front face. What shape is the cross-section?

Oblique drawing of a rectangular pyramid with a horizontal cutting plane at mid-height highlighted, connected by an arrow to an isolated cross-section shape.
3.

A right rectangular pyramid is cut with a horizontal plane parallel to the base, halfway between the base and the apex. What shape is the cross-section?

Oblique drawing of a rectangular pyramid with a vertical cutting plane through the apex highlighted, connected by an arrow to an isolated cross-section shape.
4.

A right rectangular pyramid is cut with a vertical plane that passes exactly through the apex and through the midpoints of two opposite base edges. What shape is the cross-section?

5.

A rectangular prism is sliced with a diagonal cut that is not parallel to any of its faces. The cross-section is a   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Shape name:
C

Varied Practice

1.

A rectangular prism is sliced with a horizontal plane parallel to its base. Which statement about the cross-section is always true?

Oblique drawing of a rectangular pyramid with a vertical cutting plane that misses the apex, highlighted in yellow, connected by an arrow to an isolated cross-section shape.
2.

A right rectangular pyramid is cut with a vertical plane that is parallel to one pair of base edges but does NOT pass through the apex. What shape is the cross-section?

3.

Which cutting direction applied to a right rectangular prism would produce a cross-section that is a parallelogram but NOT a rectangle?

4.

A rectangular prism is cut in two different ways: first with a horizontal plane parallel to the base, and then with a diagonal plane not parallel to any face. Describe each cross-section and explain what geometric property of the cutting plane causes the shapes to be different.

D

Word Problems

1.

A block of tofu is shaped like a right rectangular prism. A chef makes a straight horizontal cut through the middle to create two equal slabs. What shape is the exposed cut surface?

2.

A chocolate bar is shaped like a right rectangular prism. Leila cuts it with a diagonal slice that is not parallel to any of the bar's faces. What shape is the cross-section, and how do you know?

Oblique drawing of a rectangular pyramid with two labeled cuts: a dashed horizontal cut at mid-height labeled Cut 1 and a solid vertical plane through the apex labeled Cut 2.
3.

A trophy is shaped like a right rectangular pyramid. The trophy is 12 cm tall with a rectangular base. Two different cuts are made to study the shape.

1.

When the trophy is cut horizontally at its midpoint (halfway between base and apex), the cross-section is a   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Shape name:
2.

When the trophy is cut vertically through the apex (from apex straight down through the center of the base), the cross-section is a   ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲ ̲   .

Shape name:
4.

A display piece is shaped like a right rectangular pyramid. It is cut with a vertical plane that does NOT pass through the apex — the plane is parallel to one pair of triangular faces but cuts across the other two triangular faces only. The cross-section is a trapezoid. Explain why a vertical cut that DOES pass through the apex produces a triangle instead of a trapezoid.

E

Error Analysis

Two-column error contrast card. Left column (yellow) shows Maya's incorrect claim that all prism cuts produce rectangles. Right column (teal) shows the correct reasoning that diagonal cuts produce parallelograms.
1.

Maya sliced a rectangular prism with a diagonal cut that was not parallel to any face.
She wrote: "The cross-section is a rectangle because all cuts of a rectangular prism produce rectangles."

Read Maya's work below. Identify her error and choose the correct explanation.

Two-column error contrast card. Left column (yellow) shows Amir's incorrect claim that horizontal pyramid cuts produce triangles. Right column (teal) shows the correct reasoning that horizontal cuts produce rectangles.
2.

Amir sliced a right rectangular pyramid with a horizontal plane parallel to the base, cutting through the middle.
He wrote: "The cross-section is a triangle because pyramids are triangular — you can see the triangle shape when you look at a pyramid from the side."

Read Amir's work below. Identify his error and choose the correct explanation.

F

Challenge

Oblique drawing of a rectangular pyramid with three labeled cuts: horizontal at base, vertical through apex, and off-apex vertical. Three isolated cross-section shapes (rectangle, triangle, trapezoid) are shown on the right connected by dashed arrows.
1.

Three cross-sections are described: (1) a rectangle the same size as the base, (2) a triangle with its apex at the top, and (3) a trapezoid. Each cross-section came from a different cut of the same solid. Which 3D solid must this be, and what cutting direction produced each cross-section? Explain your reasoning.

2.

Compare the cross-sections of a rectangular prism and a rectangular pyramid. In what ways are they similar when cut horizontally, and in what ways do their cross-sections differ as you move the horizontal cut from the base toward the top? Explain using geometric properties.

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