Surface Area Nets

Grade 6 · geometry · 82 practice problems · read aloud

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🧊 Surface Area Nets

What is a Net and Why is it Useful?

A net is a flat, 2D shape that can be folded to form a 3D solid. It's like "unfolding" a box! Using nets helps us find the surface area—the total area of all the faces of a 3D shape. This is super useful for real-world problems, like figuring out how much wrapping paper you need for a gift box.

How to Find Surface Area Using Nets

  1. Identify the 3D Shape: What shape are you working with? (e.g., rectangular prism, cube, triangular prism).
  2. Imagine or Draw the Net: "Unfold" the shape in your mind to see all its faces laid out flat.
  3. Find the Area of Each Face: Use the correct area formula for each rectangle, square, or triangle.
  4. Add All the Areas Together: The sum is the total surface area.

Worked Examples

Example 1: Cube
A cube has side lengths of 3 cm. Its net has 6 identical squares.
Step 1: Area of one square = side × side = 3 cm × 3 cm = 9 cm².
Step 2: Total Surface Area = 6 squares × 9 cm² = 54 cm².

Example 2: Rectangular Prism
A box is 5 cm long, 4 cm wide, and 2 cm high.
Step 1: The net has 3 pairs of identical rectangles.
Step 2: Find each area:
  - Top/Bottom: 5 cm × 4 cm = 20 cm² (x2 = 40 cm²)
  - Front/Back: 5 cm × 2 cm = 10 cm² (x2 = 20 cm²)
  - Sides: 4 cm × 2 cm = 8 cm² (x2 = 16 cm²)
Step 3: Add them: 40 + 20 + 16 = 76 cm².

⚠️ Common Mistakes to Avoid

  • Forgetting Faces: Make sure you count all the faces in the net. A rectangular prism always has 6 faces!
  • Mixing Up Units: Always write the units (like cm²) for area. A number without units is incomplete.
  • Using the Wrong Formula: Remember, area of a rectangle is length × width. Don't add the sides instead of multiplying!

💡 Tips & Tricks

  • Look for Identical Faces: Find matching pairs to cut your work in half! Calculate one, then double it.
  • Label Your Net: As you draw, write the dimensions on each face so you don't get confused.
  • Check with a Model: If you're stuck, use a real box, label its faces, and then unfold it to see the net.

Practice Makes Perfect!

To get better at finding surface area with nets, try these activities:

  • Find boxes and containers around your house. Measure them and calculate how much paper would be needed to wrap them.
  • Draw nets for different 3D shapes on graph paper, cut them out, and fold them up.
  • Practice with worksheets that show a net and ask for the surface area, and vice-versa.

Practice problems

6 of the 82, worked through step by step — try them before opening the answer.

1 2³ × (15 - 7) + √144 = ?

Hint: Remember to follow the order of operations and evaluate exponents before multiplication, then combine all terms.

Show the answer

Answer: 76

  1. Handle the exponent** 2³ means 2 × 2 × 2 = 8 So now we have: 8 × (15 - 7) + √144 **
  2. Simplify inside parentheses** 15 - 7 = 8 So now: 8 × 8 + √144 **
  3. Handle the square root** √144 = 12 (since 12 × 12 = 144) So now: 8 × 8 + 12 **
  4. Perform multiplication** 8 × 8 = 64 So now: 64 + 12 **
  5. Perform addition** 64 + 12 = 76 **Final Answer:** 76

Let's solve step by step. We have: 2³ × (15 - 7) + √144 **

2 (-12)² ÷ 4 + 3 × (-5) = ?

Hint: Remember to follow the order of operations (PEMDAS) and pay attention to negative signs when squaring numbers.

Show the answer

Answer: 21

  1. Evaluate the exponent: (-12)² = 144
  2. Perform division: 144 ÷ 4 = 36
  3. Perform multiplication: 3 × (-5) = -15
  4. Add the results: 36 + (-15) = 36 - 15 = 21
  5. The final answer is 21.

3 √(169) + 4² × (18 - 9) = ?

Hint: Remember to follow the order of operations: parentheses first, then exponents, then multiplication/division from left to right, and finally addition/subtraction from left to right.

Show the answer

Answer: 157

  1. Solve inside the parentheses: 18 - 9 = 9
  2. Calculate the square root: √(169) = 13
  3. Calculate the exponent: 4² = 16
  4. Multiply: 16 × 9 = 144
  5. Add: 13 + 144 = 157

The answer is 157.

4 A rectangular prism has dimensions 8 cm × 5 cm × 12 cm. Find its surface area.

Hint: Remember that a rectangular prism has 6 faces. Opposite faces have the same area. Calculate the area of each unique face and add them all together.

Show the answer

Answer: 392 cm²

  1. Identify the three pairs of faces with dimensions: - Front and back: 8 cm × 12 cm = 96 cm² each - Left and right: 5 cm × 12 cm = 60 cm² each - Top and bottom: 8 cm × 5 cm = 40 cm² each
  2. Calculate total surface area: Front + back = 96 + 96 = 192 cm² Left + right = 60 + 60 = 120 cm² Top + bottom = 40 + 40 = 80 cm²
  3. Add all areas together: 192 + 120 + 80 = 392 cm² The surface area of the rectangular prism is 392 cm².

5 A rectangular prism has dimensions 8 cm × 5 cm × 3 cm. What is its surface area?

Hint: A rectangular prism has 6 faces. Opposite faces have equal areas. Calculate the area of each unique face type and add them together.

Show the answer

Answer: 158 cm²

  1. Identify the three pairs of faces: 8×5, 8×3, and 5×3
  2. Calculate area of 8×5 faces: 8 × 5 = 40 cm² (there are 2 of these)
  3. Calculate area of 8×3 faces: 8 × 3 = 24 cm² (there are 2 of these)
  4. Calculate area of 5×3 faces: 5 × 3 = 15 cm² (there are 2 of these)
  5. Add all areas: (40 × 2) + (24 × 2) + (15 × 2) = 80 + 48 + 30 = 158 cm²
  6. The surface area is 158 cm².

6 A rectangular prism has dimensions 14 cm × 11 cm × 9 cm. Draw its net and find the total surface area.

Hint: Think about how a net unfolds the prism into flat rectangles. There are three pairs of identical faces. Find the area of each unique rectangle, double it, and add them together.

Show the answer

Answer: 758

  1. Identify the three pairs of faces. The dimensions are length = 14 cm, width = 11 cm, height = 9 cm.
  2. Face pair 1 (front and back): 14 cm × 9 cm = 126 sq cm each. Two faces: 126 × 2 = 252 sq cm.
  3. Face pair 2 (top and bottom): 14 cm × 11 cm = 154 sq cm each. Two faces: 154 × 2 = 308 sq cm.
  4. Face pair 3 (left and right sides): 11 cm × 9 cm = 99 sq cm each. Two faces: 99 × 2 = 198 sq cm.
  5. Add all areas: 252 + 308 + 198 = 758 sq cm. The total surface area is 758 square centimeters.
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