Grade 8 Cylinder Volume Derivation: Units and Reasonableness Checks Unit Mastery Quiz with Full Solutions

Grade level and purpose

This is a Grade 8 mastery quiz for Geometry and Measurement: Volume, Cylinder Volume Derivation, Units and Reasonableness Checks.

It assesses whether you can:

• identify correct cubic units for cylinder volume,
• detect unit mistakes in work,
• decide whether an answer is reasonable from the size of the measurements,
• use the structure base area x height to justify checks,
• explain why changing a measurement changes volume the way it does.

For direct formula fluency, radius-versus-diameter practice, and broader unit review, use these linked records instead of treating this quiz as first instruction:

• rea.m08.geometry-measurement.volume.basic-formulas.cylinders.unit-mastery-quiz
• rea.m08.geometry-measurement.volume.basic-formulas.units-reasonableness.unit-mastery-quiz
• rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.unit-mastery-quiz
• rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz

Marking guidance

Recommended total: 20 marks.

Unless a question says otherwise:

• 1 mark for correct setup or reasoning,
• 1 mark for correct answer with correct units.

Use pi ~= 3.14 when a decimal is needed.

Quiz

Question 1

A cylinder has radius 4 cm and height 10 cm.

1. Write the volume with the correct units.
2. Explain why the units must be cubic units.

Question 2

A student writes:

V = pi(3 m)^2(8 m) = 226.08 m^2

Find the mistake and give the corrected answer.

Question 3

A can has radius 2.5 cm and height 12 cm.

Without calculating exactly first, decide whether the volume should be closer to:

• 20 cm^3
• 200 cm^3
• 2000 cm^3

Then calculate to confirm.

Question 4

A cylinder has diameter 10 mm and height 9 mm.

A student says the volume is pi(10)^2(9) = 900pi mm^3. Explain the error and find the correct volume.

Question 5

A water tank is a cylinder with radius 0.5 m and height 2 m.

A student gets 1.57 m^3. Is that reasonable? Explain using the size of the circular base and the height.

Question 6

Two cylinders have the same height, 7 cm.

• Cylinder A has radius 3 cm.
• Cylinder B has radius 6 cm.

Without full calculation first, predict how Cylinder B's volume compares with Cylinder A's volume. Then verify.

Question 7

A cylinder has radius 5 cm and height 11 cm. A student estimate is V ~= 25 x 11 = 275 cm^3.

1. Why is this estimate too small?
2. Give a better estimate.
3. Give the calculated volume to the nearest tenth.

Question 8

A cylinder has volume 314 cm^3 and height 10 cm. Use pi ~= 3.14 to find the radius. Then state one quick reasonableness check.

Question 9

A label says a cylindrical container has radius 4 cm, height 20 cm, and volume 100.48 cm^3. Decide whether the label is believable. Show the check.

Question 10

A cylinder-shaped column has radius 1.2 m and height 3 m. One student reports 13.6 m^3. Another reports 1.36 m^3.

Which answer is more reasonable? Show enough reasoning to justify your choice, then calculate the volume to the nearest tenth.

Full marking solutions

Solution 1

V = pi r^2 h = pi(4)^2(10) = pi(16)(10) = 160pi cm^3

Using pi ~= 3.14, V ~= 502.4 cm^3.

The units must be cubic units because volume measures three-dimensional space. In r^2 h, the units are cm^2 x cm = cm^3.

Marks:

• 1 mark for correct setup and value 160pi
• 1 mark for correct unit explanation and cm^3

Solution 2

The student's numerical calculation is fine, but the unit is wrong.

(3 m)^2 gives 9 m^2. Then multiplying by 8 m gives 72 m^3 before multiplying by pi.

So: V = pi(3)^2(8) = 72pi m^3 ~= 226.08 m^3

Corrected answer: 226.08 m^3.

Marks:

• 1 mark for identifying the unit mistake
• 1 mark for corrected answer with m^3

Solution 3

First estimate:

• r^2 = 2.5^2 = 6.25
• 6.25 x 12 = 75
• multiplying by pi gives about 75 x 3 = 225

So the volume should be closest to 200 cm^3.

Now calculate: V = pi(2.5)^2(12) = pi(6.25)(12) = 75pi cm^3 ~= 235.5 cm^3

So 200 cm^3 was the best choice.

Marks:

• 1 mark for reasonable choice with supporting estimate
• 1 mark for correct calculation

Solution 4

The error is that the student used the diameter as the radius.

If the diameter is 10 mm, then the radius is 5 mm.

Correct volume: V = pi(5)^2(9) = pi(25)(9) = 225pi mm^3

Using pi ~= 3.14: V ~= 706.5 mm^3

Marks:

• 1 mark for identifying radius-diameter error
• 1 mark for corrected volume

Solution 5

Calculate: V = pi(0.5)^2(2) = pi(0.25)(2) = 0.5pi m^3 ~= 1.57 m^3

Yes, this is reasonable.

Reasonableness check:

• The base area is pi(0.5)^2 = 0.25pi ~= 0.785 m^2
• Multiplying by height 2 m gives about 1.57 m^3
• Since the base is less than 1 m^2 and the height is 2 m, a volume a bit bigger than 1.5 m^3 makes sense.

Marks:

• 1 mark for correct calculation
• 1 mark for reasonable explanation

Solution 6

Prediction first:

• Cylinder B's radius is double Cylinder A's radius: 6 cm instead of 3 cm
• Volume depends on r^2, so doubling the radius multiplies volume by 2^2 = 4

So Cylinder B should have 4 times the volume of Cylinder A.

Verify: V_A = pi(3)^2(7) = 63pi cm^3 V_B = pi(6)^2(7) = 252pi cm^3

Check: 252pi / 63pi = 4

So the prediction is correct.

Marks:

• 1 mark for correct prediction with square-law reasoning
• 1 mark for verification

Solution 7

1. The estimate 25 x 11 = 275 is too small because it leaves out the factor pi. The base area is not 25 cm^2; it is 25pi cm^2.

2. Better estimate: 25 x 11 x 3 = 825, so a good estimate is about 825 cm^3.

3. Exact calculation: V = pi(5)^2(11) = pi(25)(11) = 275pi cm^3 ~= 863.5 cm^3

Marks:

• 1 mark for identifying missing pi
• 1 mark for improved estimate and correct calculation

Solution 8

Given: 314 = pi r^2 (10) Using pi ~= 3.14: 314 = 3.14 x r^2 x 10 314 = 31.4r^2 r^2 = 10 r ~= sqrt(10) ~= 3.16

So the radius is about 3.2 cm to the nearest tenth.

Quick reasonableness check:

• If r is a little more than 3 cm, then the base area should be a little more than pi x 9 ~= 28.3 cm^2
• Multiplying by height 10 cm gives a volume a little more than 283 cm^3
• 314 cm^3 fits that expectation, so the answer is reasonable.

Marks:

• 1 mark for solving for r
• 1 mark for a valid reasonableness check

Solution 9

Check the claimed volume: V = pi(4)^2(20) = pi(16)(20) = 320pi cm^3

Using pi ~= 3.14: V ~= 1004.8 cm^3

The label says 100.48 cm^3, which is 10 times too small.

So the label is not believable.

Marks:

• 1 mark for correct check
• 1 mark for conclusion about reasonableness

Solution 10

Estimate first:

• r^2 = 1.2^2 = 1.44
• 1.44 x 3 = 4.32
• 4.32 x pi is about 4.32 x 3.14 ~= 13.6

So 13.6 m^3 is the more reasonable answer. 1.36 m^3 is too small by a factor of 10.

Now calculate: V = pi(1.2)^2(3) = pi(1.44)(3) = 4.32pi m^3 ~= 13.6 m^3

To the nearest tenth, the volume is 13.6 m^3.

Marks:

• 1 mark for identifying the reasonable answer with estimate
• 1 mark for correct calculation

Mastery threshold recommendation

Recommend mastery at 16/20 or better, with these conditions:

• the learner must earn at least 7/10 questions substantially correct, and
• must make no more than one unit error across the quiz.

Interpretation:

• 18-20: strong mastery; ready to combine unit checks with multi-step cylinder problems.
• 16-17: secure mastery; minor slips only.
• 12-15: partial mastery; review unit tracking, radius-versus-diameter checks, and estimation.
• 0-11: not yet mastered; revisit the linked lesson and the broader volume quiz records before reassessment.

Common error patterns this quiz is designed to detect

• Writing cm^2 or m^2 for volume instead of cubic units.
• Using diameter directly as radius.
• Forgetting the factor pi when estimating or calculating.
• Treating a volume answer as reasonable without comparing it to the size of the base and the height.
• Missing that doubling the radius multiplies volume by 4, not 2.

Suggested follow-through

If a learner misses mainly unit and estimation items, return to rea.m08.geometry-measurement.volume.basic-formulas.units-reasonableness.unit-mastery-quiz.

If a learner misses formula setup or radius-diameter items, return to rea.m08.geometry-measurement.volume.basic-formulas.cylinders.unit-mastery-quiz and rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.unit-mastery-quiz.