Chapter 4 · Lesson 11 of 12 · Grade 8 · Geometry · MAT-08-GEO-011
Find Volume of Cylinders
Represent and solve find volume of cylinders problems using precise mathematical notation, multiple representations, and justified reasoning.
Math V4 · Course chapter
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry
Essential question: How can transformations and geometric relationships explain shape, distance, similarity, and measurement?
Where this lesson fits
Lesson 11 of 12. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Build and defend a multi-representation model using transformations, similarity, pythagorean geometry, and volume, including interpretation, verification, and error analysis.
Learning objectives
What you should be able to do
- Represent and solve find volume of cylinders problems using precise mathematical notation, multiple representations, and justified reasoning.
- Interpret and verify find volume of cylinders results in context, including units, constraints, and reasonableness.
Prerequisite check
Make sure the foundation is ready.
Grade 8 uses this prerequisite as active knowledge in a more formal representation.
Learn
Build the idea from meaning, not memorization.
Chapter 4: Transformations, Similarity, Pythagorean Geometry, and Volume
Geometry. Lesson 11 of 12 extends earlier chapter reasoning.
Connection to the course
Use earlier Grade 8 chapters, Grade 7 prerequisite knowledge as active knowledge. Connect today's representation to at least one earlier representation and explain the relationship.
Meaning and structure
Find Volume of Cylinders belongs to geometry. Identify the governing property or relationship before choosing a procedure.
Multiple representations
Represent find volume of cylinders numerically and symbolically, then connect it to the Grade 8 Math Box. Tables, graphs, equations, coordinate models, diagrams, and verbal descriptions must agree.
Verification and interpretation
Check with substitution, an inverse operation, an equivalent representation, geometric invariants, estimation, or context. State whether the result is exact or approximate and preserve relevant units and constraints.
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.
Solid Volume Lab
- Model find volume of cylinders in the Math Box.
- Change one input and predict which quantities or invariants should change.
- Explain what the visual, equation, and numerical result say about the same relationship.
Worked examples
Twenty different ways to see the concept work.
Find exact volume of a cylinder with r=2, h=3.
V = 12πCylinder volume is πr²h.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=4.
V = 36πA representation makes the mathematical structure visible before calculation. Cylinder volume is πr²h.
Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=5.
V = 80πJustification connects each step to a definition, property, or relationship. Cylinder volume is πr²h.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=6.
V = 150πAn estimate provides an independent reasonableness check. Cylinder volume is πr²h.
Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=7.
V = 252πIndependent verification should support the result without merely repeating the same procedure. Cylinder volume is πr²h.
Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=8.
V = 392πInterpretation connects a computed value to the mathematical situation. Cylinder volume is πr²h.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Find exact volume of a cylinder with r=8, h=9.
Rigid transformations preserve size; dilations generally do not. Correct solution: V = 576πError analysis identifies the broken idea and replaces it with valid reasoning. Cylinder volume is πr²h.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Find exact volume of a cylinder with r=9, h=10. Case B: Find exact volume of a cylinder with r=4, h=21.
Case A: V = 810π Case B: V = 336πComparison should address structure and representation, not only numerical answers. Cylinder volume is πr²h.
Find exact volume of a cylinder with r=2, h=11.
V = 44πCylinder volume is πr²h.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=12.
V = 108πA representation makes the mathematical structure visible before calculation. Cylinder volume is πr²h.
Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=13.
V = 208πJustification connects each step to a definition, property, or relationship. Cylinder volume is πr²h.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=14.
V = 350πAn estimate provides an independent reasonableness check. Cylinder volume is πr²h.
Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=15.
V = 540πIndependent verification should support the result without merely repeating the same procedure. Cylinder volume is πr²h.
Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=16.
V = 784πInterpretation connects a computed value to the mathematical situation. Cylinder volume is πr²h.
Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Find exact volume of a cylinder with r=8, h=17.
Rigid transformations preserve size; dilations generally do not. Correct solution: V = 1088πError analysis identifies the broken idea and replaces it with valid reasoning. Cylinder volume is πr²h.
Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Find exact volume of a cylinder with r=9, h=18. Case B: Find exact volume of a cylinder with r=4, h=29.
Case A: V = 1458π Case B: V = 464πComparison should address structure and representation, not only numerical answers. Cylinder volume is πr²h.
Find exact volume of a cylinder with r=2, h=19.
V = 76πCylinder volume is πr²h.
Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=20.
V = 180πA representation makes the mathematical structure visible before calculation. Cylinder volume is πr²h.
Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=21.
V = 336πJustification connects each step to a definition, property, or relationship. Cylinder volume is πr²h.
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=22.
V = 550πAn estimate provides an independent reasonableness check. Cylinder volume is πr²h.
Practice
Guided practice with hints
01Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=23.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: V = 828π
02Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=24.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: V = 1176π
03Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Find exact volume of a cylinder with r=8, h=25.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: V = 1600π
04Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Find exact volume of a cylinder with r=9, h=26. Case B: Find exact volume of a cylinder with r=4, h=37.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: Case A: V = 2106π Case B: V = 592π
05Find exact volume of a cylinder with r=2, h=27.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: V = 108π
06Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=28.
Hint: Represent the structure first, calculate carefully, then verify with a second representation or substitution.
Answer: V = 252π
Practice
Independent practice
01Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=29.
Answer: V = 464π
02Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=30.
Answer: V = 750π
03Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=31.
Answer: V = 1116π
04Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=32.
Answer: V = 1568π
05Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Find exact volume of a cylinder with r=8, h=33.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: V = 2112π
06Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Find exact volume of a cylinder with r=9, h=34. Case B: Find exact volume of a cylinder with r=4, h=45.
Answer: Case A: V = 2754π Case B: V = 720π
07Find exact volume of a cylinder with r=2, h=35.
Answer: V = 140π
08Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=36.
Answer: V = 324π
Common mistakes
Learn to catch the error, not just the answer.
Rigid transformations preserve size; dilations generally do not.
a²+b²=c² applies to right triangles.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Use transformations in design and coordinate geometry and Pythagorean reasoning for distance.
- Model capacities of cylindrical, conical, and spherical objects.
Challenge problems
Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=37.
V = 592π
Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=38.
V = 950π
Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=39.
V = 1404π
Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=40.
V = 1960π
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 8 Algebra & Modeling Sprint
Solve find volume of cylinders problems and justify each result with a valid verification method.
Practice
Mastery check
01Error analysis: a student says, "Assuming every transformation preserves size." Explain the error, then solve this related task correctly: Find exact volume of a cylinder with r=8, h=41.
Answer: Rigid transformations preserve size; dilations generally do not. Correct solution: V = 2624π
02Compare two related cases. Explain what changes and what remains mathematically consistent. Case A: Find exact volume of a cylinder with r=9, h=42. Case B: Find exact volume of a cylinder with r=4, h=53.
Answer: Case A: V = 3402π Case B: V = 848π
03Find exact volume of a cylinder with r=2, h=43.
Answer: V = 172π
04Represent the situation with a table, graph, diagram, number line, or equation before solving: Find exact volume of a cylinder with r=3, h=44.
Answer: V = 396π
05Explain why the method works, not only what steps to perform: Find exact volume of a cylinder with r=4, h=45.
Answer: V = 720π
06Estimate or predict the result first, then calculate and judge whether the answer is reasonable: Find exact volume of a cylinder with r=5, h=46.
Answer: V = 1150π
07Verify the result with substitution, an inverse operation, or a second representation: Find exact volume of a cylinder with r=6, h=47.
Answer: V = 1692π
08Interpret the result in context, including units, direction, rate, or geometric meaning: Find exact volume of a cylinder with r=7, h=48.
Answer: V = 2352π
Terminology
Words to know
- transformation
- A rule mapping a figure to an image.
- congruent
- Having the same size and shape.
- similar
- Having equal corresponding angles and proportional corresponding lengths.
- Pythagorean theorem
- For a right triangle, a²+b²=c².
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 8 scope, sequence, and public task structure
Coherent sequencing, representations, proof-oriented reasoning, and tasks.Eureka Math² Grade 8 program structure
Module coherence, cumulative practice, and algebraic fluency.enVision Mathematics Grade 8 instructional model
Problem-based entry points, visual learning, modeling, and application.Common Core State Standards for Mathematics — Grade 8
Grade-level content and mathematical-practice expectations.