Grade 8 Unit Mastery Quiz: Volume Formula Fundamentals

This assessment is for the Grade 8 topic Volume Formula Fundamentals in the geometry and measurement strand. It is intended as a mastery check after instruction, not as first exposure.

Use these linked records first for teaching and review:

• rea.m08.geometry-measurement.volume.basic-formulas.lesson for the lesson on building and using volume formulas.
• rea.m08.geometry-measurement.volume.overview for the broader unit context on prisms and cylinders.

Quiz Scope

This quiz assesses whether the learner can:

• recognize when to use V = lwh for a right rectangular prism,
• recognize when to use V = Bh or V = pi r^2 h for a cylinder,
• identify the correct dimensions from a diagram or word description,
• compute volume with correct units,
• solve reverse problems with an unknown dimension,
• distinguish linear units from square units and cubic units,
• judge whether an answer is reasonable.

Administration Notes

• Suggested time: 30 to 40 minutes.
• Calculator: optional; if used, round only when instructed.
• For questions involving pi, exact answers in terms of pi are preferred unless a decimal approximation is requested.
• Full marks require correct setup, correct substitution, accurate computation, and correct units.

Mastery Threshold Recommendation

Recommend mastery at 8/10 or better, with the additional condition that the learner must correctly solve at least 2 of the 3 reverse problems (Questions 7, 8, and 9).

Reason:

• A learner can still make one or two arithmetic slips while showing solid formula understanding.
• Reverse problems reveal whether the learner truly understands volume structure, not just button-pushing.

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Questions

1. Rectangular prism from dimensions

A right rectangular prism has length 12 cm, width 5 cm, and height 4 cm.

Find its volume.

Marks: 1

2. Rectangular prism with decimal dimensions

A storage box is a right rectangular prism with dimensions 8.5 m, 2 m, and 1.2 m.

Find its volume.

Marks: 1

3. Interpreting cubic units

A prism has volume 72 cm^3. Explain what 72 cm^3 means in terms of unit cubes.

Marks: 1

4. Cylinder from radius and height

A cylinder has radius 3 cm and height 10 cm.

Find its volume in:

1. exact form,
2. decimal form rounded to the nearest tenth.

Marks: 1

5. Cylinder from diameter and height

A soup can is shaped like a cylinder. Its diameter is 8 cm and its height is 15 cm.

Find its volume in cubic centimetres, rounded to the nearest whole number.

Marks: 1

6. Choosing the correct formula

For each solid below, write the most suitable volume formula and then find the volume.

1. A right rectangular prism with l = 9 cm, w = 2 cm, h = 7 cm
2. A cylinder with base radius 4 cm and height 6 cm

Marks: 2

7. Missing height of a rectangular prism

A right rectangular prism has volume 180 cm^3. Its length is 9 cm and its width is 4 cm.

Find the height.

Marks: 1

8. Missing radius of a cylinder

A cylinder has volume 200pi cm^3 and height 8 cm.

Find the radius.

Marks: 1

9. Missing length in a context problem

A fish tank is a right rectangular prism with volume 96 000 cm^3. Its width is 40 cm and its height is 30 cm.

Find the length.

Marks: 1

10. Error analysis

A student solves this problem:

A cylinder has diameter 10 cm and height 7 cm.

Student work: V = pi(10)^2(7) = 700pi cm^3

Identify the mistake and give the correct volume.

Marks: 1

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Full Marking Solutions

1. Rectangular prism from dimensions

Use the rectangular prism formula:

V = lwh

Substitute:

V = 12 x 5 x 4

V = 60 x 4 = 240

So the volume is:

240 cm^3

Answer: 240 cm^3

Marking note: 1 mark for correct volume with cubic units.

2. Rectangular prism with decimal dimensions

Use:

V = lwh

Substitute:

V = 8.5 x 2 x 1.2

First multiply:

8.5 x 2 = 17

Then:

17 x 1.2 = 20.4

So the volume is:

20.4 m^3

Answer: 20.4 m^3

Marking note: 1 mark for correct setup and answer with units.

3. Interpreting cubic units

72 cm^3 means the prism would hold the same space as 72 unit cubes, where each unit cube measures 1 cm x 1 cm x 1 cm.

Answer: The solid has the volume of 72 one-centimetre cubes.

Marking note: 1 mark for connecting cubic centimetres to unit cubes, not just repeating the number.

4. Cylinder from radius and height

Use the cylinder formula:

V = pi r^2 h

Substitute r = 3 and h = 10:

V = pi(3)^2(10)

V = pi(9)(10)

V = 90pi

Exact volume:

90pi cm^3

Decimal approximation:

90pi ≈ 282.7

So the volume is approximately:

282.7 cm^3

Answer: 90pi cm^3, or about 282.7 cm^3

Marking note: 1 mark for both exact and rounded decimal forms.

5. Cylinder from diameter and height

The diameter is 8 cm, so the radius is:

r = 8 / 2 = 4 cm

Now use:

V = pi r^2 h

V = pi(4)^2(15)

V = pi(16)(15)

V = 240pi

Approximate:

240pi ≈ 753.98

Rounded to the nearest whole number:

754 cm^3

Answer: 754 cm^3

Marking note: 1 mark for correctly halving the diameter and finding the volume.

6. Choosing the correct formula

6.1 Right rectangular prism

Most suitable formula:

V = lwh

Substitute:

V = 9 x 2 x 7 = 126

Volume:

126 cm^3

6.2 Cylinder

Most suitable formula:

V = pi r^2 h

Substitute:

V = pi(4)^2(6)

V = pi(16)(6) = 96pi

Volume:

96pi cm^3

Approximate decimal:

96pi ≈ 301.6 cm^3

Answers:

• Prism: V = lwh, volume 126 cm^3
• Cylinder: V = pi r^2 h, volume 96pi cm^3 or about 301.6 cm^3

Marking note:

• 1 mark for correct formula choice and correct prism volume.
• 1 mark for correct formula choice and correct cylinder volume.

7. Missing height of a rectangular prism

Use:

V = lwh

Given:

180 = 9 x 4 x h

180 = 36h

Solve for h:

h = 180 / 36 = 5

So the height is:

5 cm

Answer: 5 cm

Marking note: 1 mark for correct equation and solution.

8. Missing radius of a cylinder

Use:

V = pi r^2 h

Given:

200pi = pi r^2 (8)

Divide both sides by pi:

200 = 8r^2

Divide by 8:

25 = r^2

Take the positive square root because a radius cannot be negative:

r = 5

So the radius is:

5 cm

Answer: 5 cm

Marking note: 1 mark for correct algebra and interpretation.

9. Missing length in a context problem

Use:

V = lwh

Given:

96 000 = l x 40 x 30

Multiply the known dimensions:

40 x 30 = 1200

So:

96 000 = 1200l

Solve:

l = 96 000 / 1200 = 80

So the length is:

80 cm

Answer: 80 cm

Marking note: 1 mark for correct setup and solution.

10. Error analysis

The student used 10 cm as the radius, but 10 cm is the diameter.

So the correct radius is:

r = 10 / 2 = 5 cm

Now use the correct formula:

V = pi r^2 h

V = pi(5)^2(7)

V = pi(25)(7)

V = 175pi cm^3

Approximate decimal:

175pi ≈ 549.8 cm^3

Answer: The mistake was using the diameter instead of the radius. The correct volume is 175pi cm^3 or about 549.8 cm^3.

Marking note: 1 mark for identifying the error and giving the corrected volume.

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Scoring Guide

• Question 1: 1 mark
• Question 2: 1 mark
• Question 3: 1 mark
• Question 4: 1 mark
• Question 5: 1 mark
• Question 6: 2 marks
• Question 7: 1 mark
• Question 8: 1 mark
• Question 9: 1 mark
• Question 10: 1 mark

Total: 11 marks

For reporting as a ten-question mastery quiz, convert to a mastery judgment as follows:

• 10-11/11: strong mastery
• 8-9/11: meets mastery threshold
• 6-7/11: near mastery; targeted review needed
• 0-5/11: reteach core ideas before reassessment

Diagnostic Interpretation

Use error patterns to decide next steps.

If the learner misses Questions 1, 2, or 6.1

Likely issue: using V = lwh unreliably or multiplying dimensions carelessly.

Repair:

• rehearse identifying length, width, and height,
• estimate first to catch unreasonable products,
• restate volume as layers of cubes.

If the learner misses Questions 4, 5, 6.2, or 10

Likely issue: confusion about cylinder structure, radius versus diameter, or squaring the radius.

Repair:

• explicitly label r before substituting,
• say aloud: "square the radius, then multiply by height, then by pi",
• compare a correct and incorrect substitution side by side.

If the learner misses Questions 7, 8, or 9

Likely issue: weak reverse-problem algebra or incomplete understanding of how formulas are structured.

Repair:

• write the full formula first,
• substitute known values second,
• isolate the unknown last,
• check by substituting the found value back into the original formula.

If the learner misses Question 3

Likely issue: formulas are being memorized without understanding what volume measures.

Repair:

• return to unit-cube and layer models,
• connect cm^3 to actual 1 cm x 1 cm x 1 cm cubes,
• explain why volume counts space, not edge length or surface covering.

Common Misconceptions to Watch For

• Using diameter as radius in cylinder problems.
• Writing area units such as cm^2 instead of volume units such as cm^3.
• Forgetting to square the radius in V = pi r^2 h.
• Solving reverse problems by dividing by only one known factor instead of all known factors.
• Treating formulas as unrelated rules instead of seeing them as structured multiplication of base area and height.

Reassessment Recommendation

A learner who does not meet mastery should not simply repeat the same quiz immediately. Better sequence:

1. Review the linked lesson record.
2. Correct every missed question in writing.
3. Complete 3 to 5 fresh practice items of the missed type.
4. Reattempt a parallel quiz with the same skill balance.

This preserves the assessment function of the quiz while supporting actual learning.