Grade 8 Cylinder Volume Derivation: Units and Reasonableness Checks Practice Set

Grade 8 focus

This practice set is for learners who have already studied the linked lesson and worked examples on unit tracking and reasonableness checks for cylinder volume.

Use these companion records first if needed:

• rea.m08.geometry-measurement.volume.cylinder-derivation.units-checks.lesson
• rea.m08.geometry-measurement.volume.cylinder-derivation.units-checks.worked-examples
• rea.m08.geometry-measurement.volume.cylinder-volume-formulas.practice-set

This record does not re-teach the derivation or the full formula lesson. It concentrates on one habit: before accepting a cylinder-volume answer, check

• whether the units make sense,
• whether the setup uses radius rather than diameter,
• whether the size of the answer is reasonable,
• whether an exact answer and an approximate answer are being reported clearly.

How to use this set

For each problem:

1. Identify the measurements and their units.
2. Decide whether the expression or answer should end in cubic units.
3. Estimate mentally before or after calculating.
4. Accept or reject the result, and explain why.

Difficulty tags:

• Core = direct unit or reasonableness check
• Steady = includes comparison, explanation, or mixed-unit attention
• Stretch = multi-step error analysis or stronger justification

Practice Problems

1. Unit check only Core

A cylinder has radius 4 cm and height 9 cm. A student writes V = pi(4)^2(9) = 144pi cm^2. Is the unit correct? If not, correct it.

2. Unit check only Core

A cylinder has radius 3 m and height 10 m. Without calculating the final number, state the correct unit for volume.

3. Identify the unreasonable unit Core

Which answer could be the volume of a cylinder?

• 72 cm
• 72 cm^2
• 72 cm^3 Explain.

4. Reasonableness from rough estimation Core

A cylinder has radius 2 cm and height 5 cm. A student gets V = 6.28 cm^3. Use a quick estimate to decide whether this is reasonable.

5. Diameter versus radius check Core

A can has diameter 8 cm and height 12 cm. A student uses V = pi(8)^2(12). What mistake was made?

6. Exact versus approximate answer Core

A cylinder has radius 5 mm and height 7 mm. Write the volume in exact form and approximate form to the nearest tenth. Include units.

7. Which answer is more reasonable? Core

A cylinder has radius 10 cm and height 2 cm. Which is more reasonable for the volume: 62.8 cm^3 or 628 cm^3? Explain briefly.

8. Matching units Core

Match each measurement situation to the unit that should appear in the final answer.

• a. radius in cm, height in cm
• b. radius in m, height in m
• c. radius in mm, height in mm Choices: m^3, cm^3, mm^3

9. Check the statement Core

A student says, “Since area uses square units, volume of a cylinder should also use square units because the formula has r^2 in it.” Is the statement correct? Give a one- or two-sentence correction.

10. Estimate first Steady

A cylinder has radius 6 cm and height 10 cm. Without doing an exact calculation, decide whether about 1100 cm^3 is a reasonable volume.

11. Compare two cylinders Steady

Cylinder A: radius 3 cm, height 8 cm Cylinder B: radius 6 cm, height 8 cm A student says cylinder B has about double the volume of cylinder A because the radius doubled. Is that reasonable? Explain.

12. Spot the impossible answer Steady

A water bottle is about 25 cm tall with radius about 4 cm. Which answer is impossible, and why?

• 1256 cm^3
• 12.56 cm^3
• 1000 cm^3

13. Mixed-unit warning Steady

A cylinder has radius 5 cm and height 0.2 m. Explain why you should not immediately multiply the numbers as written. Then convert and find the volume in cm^3.

14. Error check from exact form Steady

A cylinder has radius 7 cm and height 3 cm. Student A writes V = 147pi cm^3. Student B writes V = 462pi cm^3. Which student is correct? Explain the error.

15. Reverse reasonableness check Steady

A cylinder has height 10 cm and radius 4 cm. A student reports V = 160pi cm^3 and then says this is about 160 cm^3. What is wrong with the approximation statement?

16. Choose the better estimate Steady

A cylinder has radius 9 cm and height 4 cm. Without a calculator, choose the better estimate for the volume:

• about 100 cm^3
• about 1000 cm^3 Explain.

17. Full check with calculation Stretch

A cylinder has diameter 14 cm and height 5 cm. Find the volume in exact form and approximate form to the nearest whole cubic centimetre. Then explain why your answer is reasonable.

18. Mixed units and reasonableness Stretch

A soup container has radius 6 cm and height 150 mm. Find the volume in cm^3. Then explain how you know the final size is reasonable.

19. Critique two student solutions Stretch

A cylinder has radius 2.5 m and height 4 m. Student A: V = pi(2.5)(4) = 10pi m^3 Student B: V = pi(2.5)^2(4) = 25pi m^3 Decide who is correct and explain both the calculation issue and the unit issue.

20. Find and repair the error Stretch

A student solves a problem with diameter 10 cm and height 8 cm like this: V = pi(10)^2(8) = 800pi cm^3 Find the error, repair it, and explain how a reasonableness check would catch the mistake.

21. Which cylinder description fits? Stretch

A cylinder has volume about 565 cm^3. Which description is more likely?

• a. radius 3 cm, height 20 cm
• b. radius 6 cm, height 5 cm Show enough reasoning to justify your choice.

22. Decide whether the claim is sensible Stretch

A student says, “If the radius is cut in half and the height stays the same, the volume is cut in half.” Is this true for cylinders? Explain using the formula and a quick numerical example.

23. Order of magnitude check Stretch

A storage tube has radius 0.5 m and height 2 m. A student reports the volume as 1570 m^3. Without redoing every step in detail, explain why this answer must be unreasonable, then give a reasonable approximate value.

24. Multi-step audit Stretch

A cylinder has radius 8 cm and height 11 cm. A student writes: V = pi(8)^2(11) V = pi(64)(11) V = 704pi V about 2210 cm^2 Audit the work line by line. Identify what is correct, what is incorrect, and write the fully correct final answer.

Answer Key

1. Incorrect; it should be 144pi cm^3.
2. m^3
3. 72 cm^3
4. Not reasonable; the actual volume should be around 60 cm^3, not 6.28 cm^3.
5. The student used the diameter as if it were the radius; the radius is 4 cm.
6. Exact: 175pi mm^3; approximate: 549.8 mm^3
7. 628 cm^3 is more reasonable.
8. a. cm^3, b. m^3, c. mm^3
9. Incorrect; r^2 helps create base area, but multiplying by height gives volume, so the final units are cubic.
10. Yes, reasonable; the volume is about 1130 cm^3.
11. No; doubling the radius makes the base area 4 times as large, so the volume is about 4 times as large when height stays the same.
12. 12.56 cm^3 is impossible; it is far too small for that bottle.
13. Convert first; 0.2 m = 20 cm, so V = pi(5)^2(20) = 500pi cm^3, about 1570.8 cm^3.
14. Student A is correct; Student B appears to have used diameter 14 instead of radius 7.
15. 160pi is not about 160; it is about 502.7 cm^3.
16. about 1000 cm^3
17. Exact: 245pi cm^3; approximate: 770 cm^3
18. 540pi cm^3, about 1696.5 cm^3
19. Student B is correct; Student A forgot to square the radius.
20. Corrected volume: 200pi cm^3, about 628.3 cm^3
21. b. is more likely.
22. False; halving the radius makes the volume one quarter as large if height stays the same.
23. 1570 m^3 is unreasonable; a reasonable value is about 1.57 m^3.
24. Correct exact value: 704pi cm^3; approximate value: about 2211.7 cm^3; final unit should be cm^3, not cm^2.

Full Solutions for the Hardest Third

17. Full check with calculation

Diameter 14 cm means radius 7 cm.

Use the formula: V = pi r^2 h V = pi(7)^2(5) V = pi(49)(5) V = 245pi cm^3

Approximate: 245pi ≈ 769.7 So the volume is about 770 cm^3.

Reasonableness check:

• 7^2 = 49, which is close to 50
• 50 x 5 = 250
• 250pi is a little less than 785 So an answer near 770 cm^3 makes sense.

18. Mixed units and reasonableness

The radius is 6 cm, but the height is 150 mm. Convert first. Since 10 mm = 1 cm, 150 mm = 15 cm.

Now calculate: V = pi r^2 h V = pi(6)^2(15) V = pi(36)(15) V = 540pi cm^3

Approximate: 540pi ≈ 1696.5 So the volume is about 1696.5 cm^3.

Reasonableness check:

• Base area is about 36pi ≈ 113 cm^2
• Height is 15 cm
• 113 x 15 ≈ 1695 So the result is consistent.
• Also, a container with width 12 cm and height 15 cm should hold well over 1000 cm^3, so 1696.5 cm^3 is plausible.

19. Critique two student solutions

Given r = 2.5 m and h = 4 m.

Student A wrote: V = pi(2.5)(4) = 10pi m^3 This is incorrect because the formula is V = pi r^2 h, not pi r h. The radius must be squared.

Student B wrote: V = pi(2.5)^2(4) = 25pi m^3 Check it: (2.5)^2 = 6.25 6.25 x 4 = 25 So V = 25pi m^3 is correct.

Unit check:

• m x m x m = m^3
• Volume must be in cubic metres, so m^3 is the correct final unit.

Reasonableness check:

• 25pi ≈ 78.5
• With radius 2.5 m and height 4 m, a volume around 80 m^3 is believable.
• Student A's 10pi ≈ 31.4 m^3 is too small because it ignores the second factor of the radius.

20. Find and repair the error

The cylinder has diameter 10 cm, so the radius is 5 cm. The student used 10 as the radius, which is the error.

Correct calculation: V = pi r^2 h V = pi(5)^2(8) V = pi(25)(8) V = 200pi cm^3

Approximate: 200pi ≈ 628.3 So the correct volume is about 628.3 cm^3.

Reasonableness check: If the radius were really 10 cm, the cylinder would be much wider than described. Using radius 10 gives 800pi, which is four times too large because doubling the radius multiplies r^2 by 4. That alone signals a major setup mistake.

21. Which cylinder description fits?

We compare the two choices.

Choice a: V = pi(3)^2(20) = pi(9)(20) = 180pi ≈ 565.5 cm^3

Choice b: V = pi(6)^2(5) = pi(36)(5) = 180pi ≈ 565.5 cm^3

Both descriptions are actually possible because both give about 565 cm^3.

This is a useful reasonableness lesson: different cylinders can have the same volume. A learner who only glances at the dimensions might think the taller one must have more volume, but the wider base in choice b balances the shorter height.

22. Decide whether the claim is sensible

The claim is false.

The formula is V = pi r^2 h. If the radius is cut in half, then r^2 becomes (1/2)^2 = 1/4 of the original. So the volume becomes one quarter of the original, not one half, as long as height stays the same.

Quick example:

• Original: r = 4, h = 10
• V = pi(4)^2(10) = 160pi

Half the radius:

• r = 2, h = 10
• V = pi(2)^2(10) = 40pi

And 40pi is one quarter of 160pi, not one half.

23. Order of magnitude check

Given r = 0.5 m and h = 2 m. The student says 1570 m^3.

Why this is unreasonable:

• The radius is only half a metre, so the base area is less than 1 m^2 times pi; in fact, it is about 0.25pi ≈ 0.785 m^2.
• Multiplying by height 2 m gives a volume around 1.57 m^3.
• So 1570 m^3 is about one thousand times too large.

Reasonable approximate value: V = pi(0.5)^2(2) V = pi(0.25)(2) V = 0.5pi ≈ 1.57 m^3

24. Multi-step audit

Student work: V = pi(8)^2(11) V = pi(64)(11) V = 704pi V about 2210 cm^2

Audit line by line:

• V = pi(8)^2(11) is correct. The formula and substitution are correct.
• V = pi(64)(11) is correct because 8^2 = 64.
• V = 704pi is correct because 64 x 11 = 704.
• V about 2210 cm^2 is incorrect in two ways:
  • the approximate value is slightly off if rounded to the nearest tenth or whole number with care,
  • the unit should be cm^3, not cm^2.

Correct approximation: 704pi ≈ 2211.7 So the fully correct answer is: V = 704pi cm^3 ≈ 2211.7 cm^3

Reasonableness check:

• 64 x 11 = 704
• 704 x 3 is already 2112, so a value a little above 2200 makes sense because pi is a little more than 3.

Common mistakes to watch for

• Using diameter instead of radius
• Writing square units instead of cubic units
• Forgetting to convert mixed units before substituting
• Treating a*pi as if it were just a
• Accepting a tiny answer for a large container, or a huge answer for a small one, without checking

Self-check rubric

You are ready to move on if you can usually do all of these:

• identify the correct final unit without prompting,
• estimate the size of a cylinder volume before or after calculating,
• catch diameter/radius confusion quickly,
• explain in words why an answer is unreasonable, not just say it is wrong.