Chapter 5 · Lesson 3 of 6 · Grade 6 · Geometry · MAT-06-GEO-003
Find Area of Trapezoids and Composite Polygons
Represent and solve find area of trapezoids and composite polygons 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 3 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 find area of trapezoids and composite polygons problems using precise mathematical notation, models, and reasoning.
- Explain why a method for find area of trapezoids and composite polygons 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 3 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
Find Area of Trapezoids and Composite Polygons 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 find area of trapezoids and composite polygons 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.
Find the area of a trapezoid with bases 4 and 6, height 3.
A = 1/2(4 + 6)3 = 15Average the parallel bases and multiply by perpendicular height.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 5 and 8, height 5.
A = 1/2(5 + 8)5 = 32.5The representation should show the same mathematical relationship as the calculation. Average the parallel bases and multiply by perpendicular height.
Explain why a valid method works, then solve: Find the area of a trapezoid with bases 6 and 10, height 7.
A = 1/2(6 + 10)7 = 56A complete explanation names the relationship or property being preserved. Average the parallel bases and multiply by perpendicular height.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 7 and 12, height 9.
A = 1/2(7 + 12)9 = 85.5The estimate is a reasonableness check, not a replacement for the exact result. Average the parallel bases and multiply by perpendicular height.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 8 and 14, height 3.
A = 1/2(8 + 14)3 = 33Verification should independently support the result. Average the parallel bases and multiply by perpendicular height.
Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 9 and 11, height 5.
A = 1/2(9 + 11)5 = 50State the result with its meaning, units, direction, or comparison—not only a number. Average the parallel bases and multiply by perpendicular height.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Find the area of a trapezoid with bases 10 and 13, height 7.
Label the figure and match each measure to the formula. Correct solution: A = 1/2(10 + 13)7 = 80.5Error analysis requires identifying the broken idea, not merely replacing the final answer. Average the parallel bases and multiply by perpendicular height.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the area of a trapezoid with bases 11 and 15, height 9. Case B: Find the area of a trapezoid with bases 11 and 14, height 3.
Case A: A = 1/2(11 + 15)9 = 117 Case B: A = 1/2(11 + 14)3 = 37.5Compare the structure, representation, units, and result rather than only the numbers. Average the parallel bases and multiply by perpendicular height.
Find the area of a trapezoid with bases 12 and 17, height 3.
A = 1/2(12 + 17)3 = 43.5Average the parallel bases and multiply by perpendicular height.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 4 and 10, height 5.
A = 1/2(4 + 10)5 = 35The representation should show the same mathematical relationship as the calculation. Average the parallel bases and multiply by perpendicular height.
Explain why a valid method works, then solve: Find the area of a trapezoid with bases 5 and 7, height 7.
A = 1/2(5 + 7)7 = 42A complete explanation names the relationship or property being preserved. Average the parallel bases and multiply by perpendicular height.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 6 and 9, height 9.
A = 1/2(6 + 9)9 = 67.5The estimate is a reasonableness check, not a replacement for the exact result. Average the parallel bases and multiply by perpendicular height.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 7 and 11, height 3.
A = 1/2(7 + 11)3 = 27Verification should independently support the result. Average the parallel bases and multiply by perpendicular height.
Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 8 and 13, height 5.
A = 1/2(8 + 13)5 = 52.5State the result with its meaning, units, direction, or comparison—not only a number. Average the parallel bases and multiply by perpendicular height.
Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Find the area of a trapezoid with bases 9 and 15, height 7.
Label the figure and match each measure to the formula. Correct solution: A = 1/2(9 + 15)7 = 84Error analysis requires identifying the broken idea, not merely replacing the final answer. Average the parallel bases and multiply by perpendicular height.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the area of a trapezoid with bases 10 and 12, height 9. Case B: Find the area of a trapezoid with bases 10 and 16, height 3.
Case A: A = 1/2(10 + 12)9 = 99 Case B: A = 1/2(10 + 16)3 = 39Compare the structure, representation, units, and result rather than only the numbers. Average the parallel bases and multiply by perpendicular height.
Find the area of a trapezoid with bases 11 and 14, height 3.
A = 1/2(11 + 14)3 = 37.5Average the parallel bases and multiply by perpendicular height.
Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 12 and 16, height 5.
A = 1/2(12 + 16)5 = 70The representation should show the same mathematical relationship as the calculation. Average the parallel bases and multiply by perpendicular height.
Explain why a valid method works, then solve: Find the area of a trapezoid with bases 4 and 9, height 7.
A = 1/2(4 + 9)7 = 45.5A complete explanation names the relationship or property being preserved. Average the parallel bases and multiply by perpendicular height.
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 5 and 11, height 9.
A = 1/2(5 + 11)9 = 72The estimate is a reasonableness check, not a replacement for the exact result. Average the parallel bases and multiply by perpendicular height.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 6 and 8, height 3.
Hint: Choose a representation before computing.
Answer: A = 1/2(6 + 8)3 = 21
02Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 7 and 10, height 5.
Hint: Explain what relationship or property makes your method valid.
Answer: A = 1/2(7 + 10)5 = 42.5
03Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Find the area of a trapezoid with bases 8 and 12, height 7.
Hint: Choose a representation before computing.
Answer: Label the figure and match each measure to the formula. Correct solution: A = 1/2(8 + 12)7 = 70
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the area of a trapezoid with bases 9 and 14, height 9. Case B: Find the area of a trapezoid with bases 9 and 13, height 3.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: A = 1/2(9 + 14)9 = 103.5 Case B: A = 1/2(9 + 13)3 = 33
05Find the area of a trapezoid with bases 10 and 16, height 3.
Hint: Choose a representation before computing.
Answer: A = 1/2(10 + 16)3 = 39
06Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 11 and 13, height 5.
Hint: Explain what relationship or property makes your method valid.
Answer: A = 1/2(11 + 13)5 = 60
Practice
Independent practice
01Explain why a valid method works, then solve: Find the area of a trapezoid with bases 12 and 15, height 7.
Answer: A = 1/2(12 + 15)7 = 94.5
02Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 4 and 8, height 9.
Answer: A = 1/2(4 + 8)9 = 54
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 5 and 10, height 3.
Answer: A = 1/2(5 + 10)3 = 22.5
04Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 6 and 12, height 5.
Answer: A = 1/2(6 + 12)5 = 45
05Error analysis: a student says, "Using a formula without identifying dimensions." Explain the mistake, then solve this related task correctly: Find the area of a trapezoid with bases 7 and 9, height 7.
Answer: Label the figure and match each measure to the formula. Correct solution: A = 1/2(7 + 9)7 = 56
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the area of a trapezoid with bases 8 and 11, height 9. Case B: Find the area of a trapezoid with bases 8 and 10, height 3.
Answer: Case A: A = 1/2(8 + 11)9 = 85.5 Case B: A = 1/2(8 + 10)3 = 27
07Find the area of a trapezoid with bases 9 and 13, height 3.
Answer: A = 1/2(9 + 13)3 = 33
08Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 10 and 15, height 5.
Answer: A = 1/2(10 + 15)5 = 62.5
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: Find the area of a trapezoid with bases 11 and 17, height 7.
A = 1/2(11 + 17)7 = 98
Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 12 and 14, height 9.
A = 1/2(12 + 14)9 = 117
Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 4 and 7, height 3.
A = 1/2(4 + 7)3 = 16.5
Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 5 and 9, height 5.
A = 1/2(5 + 9)5 = 35
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 5 Reasoning Lab
Solve find area of trapezoids and composite polygons 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: Find the area of a trapezoid with bases 6 and 11, height 7.
Answer: Label the figure and match each measure to the formula. Correct solution: A = 1/2(6 + 11)7 = 59.5
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Find the area of a trapezoid with bases 7 and 13, height 9. Case B: Find the area of a trapezoid with bases 7 and 12, height 3.
Answer: Case A: A = 1/2(7 + 13)9 = 90 Case B: A = 1/2(7 + 12)3 = 28.5
03Find the area of a trapezoid with bases 8 and 10, height 3.
Answer: A = 1/2(8 + 10)3 = 27
04Represent before calculating. Use a labeled diagram, decomposition, net, coordinate model, or formula for this task, then solve: Find the area of a trapezoid with bases 9 and 12, height 5.
Answer: A = 1/2(9 + 12)5 = 52.5
05Explain why a valid method works, then solve: Find the area of a trapezoid with bases 10 and 14, height 7.
Answer: A = 1/2(10 + 14)7 = 84
06Estimate or predict first, then calculate and decide whether the result is reasonable: Find the area of a trapezoid with bases 11 and 16, height 9.
Answer: A = 1/2(11 + 16)9 = 121.5
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Find the area of a trapezoid with bases 12 and 18, height 3.
Answer: A = 1/2(12 + 18)3 = 45
08Interpret the answer in context after solving. What does the result mean here? Find the area of a trapezoid with bases 4 and 6, height 5.
Answer: A = 1/2(4 + 6)5 = 25
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.