Introduction
Cross-Sections of Three-Dimensional Figures is an important Grade 7 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with cross-sections of three-dimensional figures.
What Is Cross-Sections of Three-Dimensional Figures?
Cross-Sections of Three-Dimensional Figures means choosing a model, naming what each number means, and explaining the strategy.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Cross-Sections of Three-Dimensional Figures
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the question carefully and identify what is being asked.
- Choose a model, equation, table, or diagram that matches the situation.
- Solve one step at a time and keep units or labels attached.
- Use the answer explanation to check that the result makes sense.
Visual Models
Visual Model 1
Question: A plane cuts diagonally through the cube as shown. What is the shape of the resulting cross-section?
- A. Rectangle
- B. Square
- C. Parallelogram
- D. Isosceles trapezoid
Why it works: A plane through opposite edges of a cube forms a rectangular cross-section. One pair of sides equals the edge length; the other pair equals the face diagonal (\(a\sqrt{2}\) for edge \(a\)).
Answer: Rectangle
Visual Model 2
Question: A cylinder is sliced by an oblique plane, as shown. What is the shape of the resulting cross-section?
- A. Circle
- B. Rectangle
- C. Ellipse
- D. Parabola
Why it works: When a plane cuts through a cylinder at an angle (not parallel to the bases and not perpendicular to the axis), the cross-section is an ellipse.
Answer: Ellipse
Worked Examples
Example 1
Question: A cone is cut by a plane perpendicular to its axis, as shown. What is the shape of the resulting cross-section?
- A. Triangle
- B. Ellipse
- C. Circle
- D. Rectangle
- Any plane perpendicular to a cone's axis creates a circular cross-section.
- The smaller the radius of the cross-section, the higher up on the cone the cut is made.
Answer: Circle
Example 2
Question: A square pyramid is cut by a plane parallel to its base. The cross-section is a square. If the pyramid has a height of 12 cm and the cut is made 4 cm from the apex, how does the area of the cross-section compare to the base?
- A. It is \(\frac{1}{3}\) of the base area
- B. It is \(\frac{1}{9}\) of the base area
- C. It is \(\frac{4}{9}\) of the base area
- D. It is \(\frac{4}{12}\) of the base area
- The ratio of areas equals the square of the ratio of corresponding lengths.
- The cut is at height 4 cm out of 12 cm, so the ratio is \(\frac{4}{12} = \frac{1}{3}\).
- Squaring: \((\frac{1}{3})^2 = \frac{1}{9}\).
Answer: It is \(\frac{1}{9}\) of the base area
Example 3
Question: A cube is sliced by a vertical plane parallel to one of its faces, passing through the center. What is the area of the cross-section if the cube has a side length of 4 cm?
- A. \(8\text{ cm}^2\)
- B. \(16\text{ cm}^2\)
- C. \(32\text{ cm}^2\)
- D. \(64\text{ cm}^2\)
- A plane parallel to a cube's face produces a square cross-section with side length 4 cm.
- The area is \(4 \times 4 = 16\text{ cm}^2\).
Answer: \(16\text{ cm}^2\)
Real-World Word Problems
Problem 1
Question: What is the shape of the cross-section formed when a cube is sliced parallel to one of its faces?
- A. Rectangle (but not square)
- B. Rhombus
- C. Square
- D. Parallelogram (but not square)
Why it works: A cube has all square faces. A cut parallel to any face produces a square cross-section congruent to that face.
Answer: Square
Problem 2
Question: A rectangular prism has dimensions \(4\text{ cm} \times 6\text{ cm} \times 8\text{ cm}\). If a vertical plane cuts through it perpendicular to the base, which shape MUST appear in the cross-section?
- A. Square
- B. Rectangle
- C. Trapezoid
- D. Triangle
Why it works: Any vertical plane perpendicular to the base of a rectangular prism intersects the solid in a rectangular cross-section (width is one base dimension, height is the prism height).
Answer: Rectangle
Common Mistakes
- Rushing before identifying what the numbers represent.
- Choosing an operation that does not match the situation.
- Dropping labels, units, or context from the answer.
- Skipping the estimate or reasonableness check.
Strategy Tips
- Underline the question being asked.
- Use a model before jumping to computation.
- Write an equation that matches the story or picture.
- Explain the final answer in a sentence.
Practice Questions
Question 1
When a cylinder is cut by a horizontal plane parallel to its circular bases, what is the shape of the cross-section?
- A. Rectangle
- B. Ellipse
- C. Circle
- D. Annulus (ring shape)
Question 2
Which of the following statements is true about cross-sections of a cube?
- A. All cross-sections are squares
- B. A cross-section can be a triangle, rectangle, or hexagon
- C. Cross-sections are always congruent to the cube's faces
- D. No cross-section can have more than 4 sides
Question 3
A cone is sliced horizontally (parallel to its circular base). What is the shape of the cross-section?
- A. Rectangle
- B. Triangle
- C. Circle
- D. Ellipse
Question 4
A pyramid with a square base is sliced parallel to its base. Which statement about the cross-section is correct?
- A. The cross-section is a rectangle
- B. The cross-section is a square, smaller than the base
- C. The cross-section is a triangle
- D. The cross-section is an irregular hexagon
Question 5
A rectangular prism with dimensions \(5\text{ cm} \times 7\text{ cm} \times 10\text{ cm}\) is cut by a plane parallel to the base. The cross-section is a rectangle with area \(35\text{ cm}^2\). Which dimension of the prism determines the area of the cross-section?
- A. The height only
- B. The base dimensions (\(5\text{ cm} \times 7\text{ cm}\))
- C. The sum of all dimensions
- D. The lateral surface area
Question 6
A right rectangular prism has dimensions \(5 \text{ cm} \times 7 \text{ cm} \times 12 \text{ cm}\). A plane cuts it at an angle, passing through one edge and an opposite edge. Which shape is LEAST likely as a cross-section?
- A. Triangle
- B. Pentagon
- C. Hexagon
- D. Rectangle
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: Circle
A horizontal plane parallel to the bases of a cylinder creates a circular cross-section congruent to the bases. The radius of this circle equals the radius of the cylinder.
Question 2
Answer: A cross-section can be a triangle, rectangle, or hexagon
Depending on the angle and position of the cutting plane, a cube's cross-section may have 3, 4, 5, or 6 sides. A triangle results from a corner cut; rectangles from parallel cuts; hexagons from diagonal cuts at certain angles.
Question 3
Answer: Circle
A horizontal plane parallel to the base of a cone produces a circular cross-section smaller than the original base. The circle is similar to the base but scaled down based on the height of the cut.
Question 4
Answer: The cross-section is a square, smaller than the base
A square pyramid cut parallel to its base produces a square cross-section. This square is similar to the base but smaller in size, with the side length proportional to the distance from the apex.
Question 5
Answer: The base dimensions (\(5\text{ cm} \times 7\text{ cm}\))
A plane parallel to the base always produces a cross-section congruent to the base. The cross-section's area is \(5 \times 7 = 35\text{ cm}^2\), independent of the prism's height.
Question 6
Answer: Hexagon
A rectangular prism has 6 faces. A plane through opposite edges typically intersects 4 or 5 faces, producing a quadrilateral or pentagon. A hexagon would require intersecting all 6 faces, which a plane through opposite edges cannot do.
Connection to Standards
This lesson supports Grade 7 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Cross-Sections of Three-Dimensional Figures becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Understand the model before choosing the operation.
