3D Cross Sections

Grade 6 · geometry · 100 practice problems · read aloud

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What is a 3D Cross Section?

A cross section is the flat, 2D shape you get when you slice straight through a 3D object. Imagine slicing a piece of cheese or an apple—the face of that slice is the cross section! 🧀

Why is it useful? It helps us understand what 3D objects look like on the inside. Architects and doctors use this idea to design buildings and read MRI scans!

How to Find a Cross Section: Step-by-Step

  1. Identify the 3D Solid: What shape are you slicing? (e.g., cube, cylinder, pyramid)
  2. Identify the Slicing Plane: Imagine a flat sheet cutting through the solid. Is it a vertical, horizontal, or angled slice?
  3. Visualize the Slice: What 2D shape is created where the plane meets the solid? Trace the edges with your mind.
  4. Name the 2D Shape: Is it a square, rectangle, circle, or triangle?

Visual Examples

Example 1: Slicing a Cube

Scenario: You slice a cube (like a dice) straight down, parallel to one of its faces.

Cross Section: You get a square! All sides are equal.

Example 2: Slicing a Rectangular Prism

Scenario: You slice a rectangular prism (like a shoebox) straight across, parallel to its largest face.

Cross Section: You get a rectangle.

Example 3: Slicing a Cylinder

Scenario: You slice a cylinder (like a soup can) horizontally, parallel to its circular base.

Cross Section: You get a circle.

Different Slice: If you slice it vertically through the center, you get a rectangle!

Common Mistakes to Avoid

Mistake 1: Thinking the cross section is always the same as the base.

Fix: Remember, the slice's direction changes everything! A vertical slice through a cylinder gives a rectangle, not a circle.

Mistake 2: Not visualizing the slice clearly.

Fix: Use clay or play-dough to make the 3D shape and actually slice it with a plastic knife to see the cross section.

Tips & Tricks

Memory Aid: "The slice tells the story!" The angle and direction of your cut determine the 2D shape you'll find.

Strategy: For tricky shapes, try drawing the 3D object and sketching the slicing plane. Then, trace the outline of the face that is created.

How to Practice

  • Use modeling clay to create different 3D shapes (cubes, pyramids, cones) and slice them in various ways with string or a plastic knife.
  • Look for real-world examples: What shape is a slice of bread? A slice of carrot? A slice of a wedding cake?
  • Draw the cross section you would get from slicing a sphere (like a melon) in half. What shape is it?

Practice problems

6 of the 100, worked through step by step — try them before opening the answer.

1 (3/4) ÷ (2/5) = ?

Hint: When dividing fractions, remember to multiply by the reciprocal of the second fraction.

Show the answer

Answer: 15/8

  1. Understand that dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 2/5 is 5/2.
  2. Rewrite the division as multiplication: (3/4) ÷ (2/5) = (3/4) × (5/2)
  3. Multiply the numerators: 3 × 5 = 15
  4. Multiply the denominators: 4 × 2 = 8
  5. Combine into a fraction: 15/8
  6. Check if it can be simplified: 15 and 8 have no common factors other than 1, so 15/8 is already in simplest form. Final answer: 15/8

We are dividing two fractions: (3/4) ÷ (2/5).

2 (-12 + 8) × (-3) = ?

Hint: Remember to perform operations inside parentheses first, then consider how multiplying negative numbers works.

Show the answer

Answer: 12

  1. First, handle the parentheses: (-12 + 8) -12 + 8 = -4 So now we have: (-4) × (-3)
  2. Multiply the two numbers: (-4) × (-3) When multiplying two negative numbers, the result is positive. 4 × 3 = 12 So (-4) × (-3) = 12
  3. Final answer: 12

Let's solve step-by-step.

3 (-15) × 4 ÷ (-3) = ?

Hint: Remember the rules for multiplying and dividing negative numbers. Work step by step from left to right.

Show the answer

Answer: 20

  1. Write the expression clearly. (-15) × 4 ÷ (-3)
  2. Perform multiplication first (since multiplication and division have the same priority, we go left to right). (-15) × 4 = -60 Now the expression becomes: -60 ÷ (-3)
  3. Divide -60 by -3. Dividing two negative numbers gives a positive result. -60 ÷ (-3) = 60 ÷ 3 = 20
  4. Final answer. 20

Let's solve step-by-step:

4 (-12) + 7 - (-5) = ?

Hint: Remember that subtracting a negative number is the same as adding its positive value. Work through the operations from left to right.

Show the answer

Answer: 0

  1. Write the original problem (-12) + 7 - (-5)
  2. Interpret the subtraction of a negative number Subtracting a negative number is the same as adding its positive: - (-5) becomes + 5 So now we have: (-12) + 7 + 5
  3. Add the numbers from left to right First: (-12) + 7 Think of this as 7 - 12 = -5 So we get: -5
  4. Now add the last number -5 + 5 = 0
  5. Final answer 0

Let's solve step-by-step:

5 (-15) + 28 - (-7) = ?

Hint: Remember that subtracting a negative number is the same as adding its positive value. Work through the operations from left to right.

Show the answer

Answer: 20

  1. Write the problem clearly (-15) + 28 - (-7)
  2. Interpret the subtraction of a negative number Subtracting a negative is the same as adding a positive: - (-7) = + 7 So the expression becomes: (-15) + 28 + 7
  3. Add the numbers from left to right First: (-15) + 28 Think of it as 28 - 15 = 13 So now we have: 13 + 7
  4. Add the last number 13 + 7 = 20 Final Answer: 20

Let's solve step-by-step:

6 (-15) + 27 - (-8) = ?

Hint: When working with negative numbers, remember that subtracting a negative is the same as adding a positive. Consider using a number line to visualize the operations.

Show the answer

Answer: 20

  1. Write the original problem (-15) + 27 - (-8)
  2. Interpret the subtraction of a negative number Subtracting a negative number is the same as adding its positive: - (-8) becomes + 8 So now the expression is: (-15) + 27 + 8
  3. Add the numbers from left to right First: (-15) + 27 Think of this as 27 - 15 = 12 So we get: 12
  4. Now add 8 to 12 12 + 8 = 20
  5. Final answer 20

Let's solve step-by-step:

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