Chapter 5 · Lesson 4 of 6 · Grade 6 · Geometry · MAT-06-GEO-006
Build Nets for Three-Dimensional Figures
Represent and solve build nets for three-dimensional figures problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 5: Area, Surface Area, and Volume
Measuring two- and three-dimensional space
Essential question: Why do geometric formulas work, and how can decomposition, nets, and unit structure justify them?
Where this lesson fits
Lesson 4 of 6. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Design packaging for an object, compare possible nets, calculate material area and interior volume, and justify the chosen design.
Learning objectives
What you should be able to do
- Represent and solve build nets for three-dimensional figures problems using precise mathematical notation, models, and reasoning.
- Explain why a method for build nets for three-dimensional figures works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Chapter 5: Area, Surface Area, and Volume
Measuring two- and three-dimensional space. This is Lesson 4 of 6 in Chapter 5. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Chapter 2 fraction operations, Chapter 3 coordinate reasoning, Chapter 4 expressions. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Build Nets for Three-Dimensional Figures connects measurement to structure. Formulas are justified by decomposition, rearrangement, similarity, coordinate reasoning, or invariance.
Represent the relationship
The Math Box uses geometry workspace to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Geometry Explorer
- Model one example of build nets for three dimensional figures in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
A prism measures 2 by 3 by 4. List the three face-size pairs in its net.
2×3, 2×4, 3×4; each appears twiceOpposite faces of a rectangular prism are congruent.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 3 by 4 by 6. List the three face-size pairs in its net.
3×4, 3×6, 4×6; each appears twiceThe representation should show the same mathematical relationship as the calculation. Opposite faces of a rectangular prism are congruent.
Explain why a valid method works, then solve: A prism measures 4 by 5 by 8. List the three face-size pairs in its net.
4×5, 4×8, 5×8; each appears twiceA complete explanation names the relationship or property being preserved. Opposite faces of a rectangular prism are congruent.
Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 5 by 6 by 10. List the three face-size pairs in its net.
5×6, 5×10, 6×10; each appears twiceThe estimate is a reasonableness check, not a replacement for the exact result. Opposite faces of a rectangular prism are congruent.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 6 by 7 by 5. List the three face-size pairs in its net.
6×7, 6×5, 7×5; each appears twiceVerification should independently support the result. Opposite faces of a rectangular prism are congruent.
Interpret the answer in context after solving. What does the result mean here? A prism measures 7 by 3 by 7. List the three face-size pairs in its net.
7×3, 7×7, 3×7; each appears twiceState the result with its meaning, units, direction, or comparison—not only a number. Opposite faces of a rectangular prism are congruent.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A prism measures 8 by 4 by 9. List the three face-size pairs in its net.
Label the figure and match each measure to the formula. Correct solution: 8×4, 8×9, 4×9; each appears twiceError analysis requires identifying the broken idea, not merely replacing the final answer. Opposite faces of a rectangular prism are congruent.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A prism measures 9 by 5 by 4. List the three face-size pairs in its net. Case B: A prism measures 18 by 4 by 8. List the three face-size pairs in its net.
Case A: 9×5, 9×4, 5×4; each appears twice Case B: 18×4, 18×8, 4×8; each appears twiceCompare the structure, representation, units, and result rather than only the numbers. Opposite faces of a rectangular prism are congruent.
A prism measures 10 by 6 by 6. List the three face-size pairs in its net.
10×6, 10×6, 6×6; each appears twiceOpposite faces of a rectangular prism are congruent.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 11 by 7 by 8. List the three face-size pairs in its net.
11×7, 11×8, 7×8; each appears twiceThe representation should show the same mathematical relationship as the calculation. Opposite faces of a rectangular prism are congruent.
Explain why a valid method works, then solve: A prism measures 12 by 3 by 10. List the three face-size pairs in its net.
12×3, 12×10, 3×10; each appears twiceA complete explanation names the relationship or property being preserved. Opposite faces of a rectangular prism are congruent.
Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 13 by 4 by 5. List the three face-size pairs in its net.
13×4, 13×5, 4×5; each appears twiceThe estimate is a reasonableness check, not a replacement for the exact result. Opposite faces of a rectangular prism are congruent.
Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 14 by 5 by 7. List the three face-size pairs in its net.
14×5, 14×7, 5×7; each appears twiceVerification should independently support the result. Opposite faces of a rectangular prism are congruent.
Interpret the answer in context after solving. What does the result mean here? A prism measures 15 by 6 by 9. List the three face-size pairs in its net.
15×6, 15×9, 6×9; each appears twiceState the result with its meaning, units, direction, or comparison—not only a number. Opposite faces of a rectangular prism are congruent.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A prism measures 16 by 7 by 4. List the three face-size pairs in its net.
Label the figure and match each measure to the formula. Correct solution: 16×7, 16×4, 7×4; each appears twiceError analysis requires identifying the broken idea, not merely replacing the final answer. Opposite faces of a rectangular prism are congruent.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A prism measures 17 by 3 by 6. List the three face-size pairs in its net. Case B: A prism measures 26 by 7 by 10. List the three face-size pairs in its net.
Case A: 17×3, 17×6, 3×6; each appears twice Case B: 26×7, 26×10, 7×10; each appears twiceCompare the structure, representation, units, and result rather than only the numbers. Opposite faces of a rectangular prism are congruent.
A prism measures 18 by 4 by 8. List the three face-size pairs in its net.
18×4, 18×8, 4×8; each appears twiceOpposite faces of a rectangular prism are congruent.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 19 by 5 by 10. List the three face-size pairs in its net.
19×5, 19×10, 5×10; each appears twiceThe representation should show the same mathematical relationship as the calculation. Opposite faces of a rectangular prism are congruent.
Explain why a valid method works, then solve: A prism measures 20 by 6 by 5. List the three face-size pairs in its net.
20×6, 20×5, 6×5; each appears twiceA complete explanation names the relationship or property being preserved. Opposite faces of a rectangular prism are congruent.
Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 21 by 7 by 7. List the three face-size pairs in its net.
21×7, 21×7, 7×7; each appears twiceThe estimate is a reasonableness check, not a replacement for the exact result. Opposite faces of a rectangular prism are congruent.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 22 by 3 by 9. List the three face-size pairs in its net.
Hint: Choose a representation before computing.
Answer: 22×3, 22×9, 3×9; each appears twice
02Interpret the answer in context after solving. What does the result mean here? A prism measures 23 by 4 by 4. List the three face-size pairs in its net.
Hint: Explain what relationship or property makes your method valid.
Answer: 23×4, 23×4, 4×4; each appears twice
03Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A prism measures 24 by 5 by 6. List the three face-size pairs in its net.
Hint: Choose a representation before computing.
Answer: Label the figure and match each measure to the formula. Correct solution: 24×5, 24×6, 5×6; each appears twice
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A prism measures 25 by 6 by 8. List the three face-size pairs in its net. Case B: A prism measures 34 by 5 by 5. List the three face-size pairs in its net.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: 25×6, 25×8, 6×8; each appears twice Case B: 34×5, 34×5, 5×5; each appears twice
05A prism measures 26 by 7 by 10. List the three face-size pairs in its net.
Hint: Choose a representation before computing.
Answer: 26×7, 26×10, 7×10; each appears twice
06Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 27 by 3 by 5. List the three face-size pairs in its net.
Hint: Explain what relationship or property makes your method valid.
Answer: 27×3, 27×5, 3×5; each appears twice
Practice
Independent practice
01Explain why a valid method works, then solve: A prism measures 28 by 4 by 7. List the three face-size pairs in its net.
Answer: 28×4, 28×7, 4×7; each appears twice
02Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 29 by 5 by 9. List the three face-size pairs in its net.
Answer: 29×5, 29×9, 5×9; each appears twice
03Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 30 by 6 by 4. List the three face-size pairs in its net.
Answer: 30×6, 30×4, 6×4; each appears twice
04Interpret the answer in context after solving. What does the result mean here? A prism measures 31 by 7 by 6. List the three face-size pairs in its net.
Answer: 31×7, 31×6, 7×6; each appears twice
05Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A prism measures 32 by 3 by 8. List the three face-size pairs in its net.
Answer: Label the figure and match each measure to the formula. Correct solution: 32×3, 32×8, 3×8; each appears twice
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A prism measures 33 by 4 by 10. List the three face-size pairs in its net. Case B: A prism measures 42 by 3 by 7. List the three face-size pairs in its net.
Answer: Case A: 33×4, 33×10, 4×10; each appears twice Case B: 42×3, 42×7, 3×7; each appears twice
07A prism measures 34 by 5 by 5. List the three face-size pairs in its net.
Answer: 34×5, 34×5, 5×5; each appears twice
08Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 35 by 6 by 7. List the three face-size pairs in its net.
Answer: 35×6, 35×7, 6×7; each appears twice
Common mistakes
Learn to catch the error, not just the answer.
Label the figure and match each measure to the formula.
Use square units for area and cubic units for volume.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Estimate material, floor space, packaging, distance, surface coverage, and capacity.
- Use diagrams and coordinates to justify measurements rather than relying on appearance.
Challenge problems
Explain why a valid method works, then solve: A prism measures 36 by 7 by 9. List the three face-size pairs in its net.
36×7, 36×9, 7×9; each appears twice
Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 37 by 3 by 4. List the three face-size pairs in its net.
37×3, 37×4, 3×4; each appears twice
Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 38 by 4 by 6. List the three face-size pairs in its net.
38×4, 38×6, 4×6; each appears twice
Interpret the answer in context after solving. What does the result mean here? A prism measures 39 by 5 by 8. List the three face-size pairs in its net.
39×5, 39×8, 5×8; each appears twice
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 5 Reasoning Lab
Solve build nets for three-dimensional figures problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: A prism measures 40 by 6 by 10. List the three face-size pairs in its net.
Answer: Label the figure and match each measure to the formula. Correct solution: 40×6, 40×10, 6×10; each appears twice
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: A prism measures 41 by 7 by 5. List the three face-size pairs in its net. Case B: A prism measures 50 by 6 by 9. List the three face-size pairs in its net.
Answer: Case A: 41×7, 41×5, 7×5; each appears twice Case B: 50×6, 50×9, 6×9; each appears twice
03A prism measures 42 by 3 by 7. List the three face-size pairs in its net.
Answer: 42×3, 42×7, 3×7; each appears twice
04Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: A prism measures 43 by 4 by 9. List the three face-size pairs in its net.
Answer: 43×4, 43×9, 4×9; each appears twice
05Explain why a valid method works, then solve: A prism measures 44 by 5 by 4. List the three face-size pairs in its net.
Answer: 44×5, 44×4, 5×4; each appears twice
06Estimate or predict first, then calculate and decide whether the result is reasonable: A prism measures 45 by 6 by 6. List the three face-size pairs in its net.
Answer: 45×6, 45×6, 6×6; each appears twice
07Solve and verify the result with a second method, inverse operation, or equivalent representation: A prism measures 46 by 7 by 8. List the three face-size pairs in its net.
Answer: 46×7, 46×8, 7×8; each appears twice
08Interpret the answer in context after solving. What does the result mean here? A prism measures 47 by 3 by 10. List the three face-size pairs in its net.
Answer: 47×3, 47×10, 3×10; each appears twice
Terminology
Words to know
- dimension
- A measurable extent such as length, width, or height.
- area
- Two-dimensional measure in square units.
- surface area
- Total area of exterior faces or surfaces.
- volume
- Three-dimensional space measured in cubic units.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.