Grade 8 Building the Cylinder Volume Formula: Unit Mastery Quiz with Full Solutions

Grade level and focus

This record is for Grade 8 Geometry and Measurement. It is a unit mastery quiz for the topic Building the Volume Formula within cylinder volume derivation.

This quiz does not reteach the full lesson. It checks whether you can:

• connect a cylinder to the prism rule V = Bh
• identify the base area of a cylinder as pi r^2
• justify why V = pi r^2 h
• interpret radius, diameter, and perpendicular height correctly
• solve direct and reverse problems while keeping units consistent

Use with linked records

Use this assessment alongside, not instead of, the related quizzes:

• Broader volume fluency: rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz
• Cylinder formula practice: rea.m08.geometry-measurement.volume.basic-formulas.cylinders.unit-mastery-quiz
• Choosing the right formula: rea.m08.geometry-measurement.volume.basic-formulas.formula-selection.unit-mastery-quiz
• Prism foundation V = Bh: rea.m08.geometry-measurement.volume.basic-formulas.prism-bh.unit-mastery-quiz

Quiz instructions

• Show all reasoning.
• Use exact answers with pi unless a decimal is requested.
• Total: 20 marks
• Suggested time: 25-30 minutes

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Questions

1. From prism rule to cylinder rule (2 marks)

A prism has volume V = Bh, where B is the area of the base.

Explain how this rule leads to the cylinder formula V = pi r^2 h.

2. Identifying the base area (2 marks)

A cylinder has radius 5 cm.

Find the area of one circular base.

3. Building the full formula for a specific cylinder (2 marks)

A cylinder has radius 4 m and height 9 m.

Write a volume expression for the cylinder and simplify it.

4. Meaning of each part of the formula (2 marks)

In the formula V = pi r^2 h, state what each of the following represents:

• r
• pi r^2
• h
• the whole product pi r^2 h

5. Radius or diameter? (2 marks)

A cylinder has diameter 12 cm and height 10 cm.

Find its volume.

6. Why height must be perpendicular (2 marks)

A student says, "Any side length of the cylinder can be used for h because height just means how tall the shape looks."

Explain what is wrong with this statement.

7. Direct volume calculation with units (2 marks)

A can is shaped like a cylinder with radius 3.5 cm and height 8 cm.

Find the volume in terms of pi, then as a decimal to the nearest tenth.

8. Reverse problem: solving for height (2 marks)

A cylinder has volume 200pi cm^3 and radius 5 cm.

Find the height.

9. Comparing two cylinders structurally (2 marks)

Cylinder A and Cylinder B have the same height.

• Cylinder A has radius 2 cm.
• Cylinder B has radius 4 cm.

Without calculating decimal volumes, determine how many times Cylinder B's volume is compared with Cylinder A's volume. Explain.

10. Error analysis (2 marks)

A student solves a problem for a cylinder with radius 6 cm and height 11 cm like this:

V = 2pi rh = 2pi(6)(11) = 132pi cm^3

Identify the mistake and give the correct volume.

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Full marking solutions

1. From prism rule to cylinder rule

A cylinder is like a prism in the sense that it has a constant cross-section all the way through, so its volume can also be found by:

V = Bh

For a cylinder, the base is a circle, so:

B = pi r^2

Substitute that into V = Bh:

V = (pi r^2)h = pi r^2 h

Marking:

• 1 mark for stating V = Bh and that a cylinder uses the same structure
• 1 mark for replacing B with pi r^2 and obtaining V = pi r^2 h

2. Identifying the base area

A = pi r^2

A = pi(5)^2

A = 25pi cm^2

Answer: 25pi cm^2

Marking:

• 1 mark for correct substitution
• 1 mark for correct simplified area with units

3. Building the full formula for a specific cylinder

V = pi r^2 h

V = pi(4)^2(9)

V = pi(16)(9)

V = 144pi m^3

Answer: 144pi m^3

Marking:

• 1 mark for correct formula setup
• 1 mark for correct simplification and units

4. Meaning of each part of the formula

• r is the radius of the circular base
• pi r^2 is the area of the circular base
• h is the perpendicular height of the cylinder
• pi r^2 h is the volume of the cylinder

Marking:

• 2 marks for four correct meanings; 1 mark if two or three are correct

5. Radius or diameter?

Diameter is 12 cm, so radius is:

r = 12 / 2 = 6 cm

Now use the formula:

V = pi r^2 h

V = pi(6)^2(10)

V = pi(36)(10)

V = 360pi cm^3

Answer: 360pi cm^3

Marking:

• 1 mark for converting diameter to radius
• 1 mark for correct volume

6. Why height must be perpendicular

The statement is wrong because in volume formulas, h means the perpendicular distance from one base to the other, not just any slanted or visible side measurement. The formula V = Bh multiplies the base area by how far the solid extends straight from the base, so using a non-perpendicular length would give the wrong volume.

Marking:

• 1 mark for identifying that height must be perpendicular
• 1 mark for explaining why this matters in V = Bh

7. Direct volume calculation with units

V = pi r^2 h

V = pi(3.5)^2(8)

V = pi(12.25)(8)

V = 98pi cm^3

Decimal approximation:

98pi ≈ 307.9

Answer: 98pi cm^3, or about 307.9 cm^3

Marking:

• 1 mark for exact answer 98pi cm^3
• 1 mark for correct decimal to the nearest tenth

8. Reverse problem: solving for height

Given:

• V = 200pi cm^3
• r = 5 cm

Use V = pi r^2 h:

200pi = pi(5)^2 h

200pi = 25pi h

Divide both sides by 25pi:

h = 8

Answer: 8 cm

Marking:

• 1 mark for correct equation/substitution
• 1 mark for correct height

9. Comparing two cylinders structurally

Volume depends on r^2 when height is unchanged:

V = pi r^2 h

Compare the radii:

• Cylinder A: r = 2
• Cylinder B: r = 4

Since 4 is 2 times 2, the squared factor is:

(4^2) / (2^2) = 16 / 4 = 4

So Cylinder B has 4 times the volume of Cylinder A.

Answer: Cylinder B has 4 times the volume.

Marking:

• 1 mark for using the squared-radius idea
• 1 mark for correct comparison result

10. Error analysis

The student used 2pi rh, which is not the cylinder volume formula. That expression comes from the circle circumference formula 2pi r, not base area.

For volume, use:

V = pi r^2 h

V = pi(6)^2(11)

V = pi(36)(11)

V = 396pi cm^3

Answer: The mistake is using circumference instead of base area. The correct volume is 396pi cm^3.

Marking:

• 1 mark for identifying the conceptual mistake
• 1 mark for correct volume

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Mastery threshold recommendation

Recommend mastery at 16/20 or higher, with Question 1 and Question 6 both substantially correct.

Why this threshold is appropriate:

• a learner should not only compute, but also understand how the formula is built
• Questions 1 and 6 test the structural ideas behind the formula, not just substitution
• a learner who scores well but misses the derivation meaning should revisit the lesson before moving on

Quick interpretation guide

• 18-20: Strong mastery of derivation, structure, and use
• 16-17: Secure mastery; ready to move to broader mixed volume work
• 13-15: Partial mastery; review the prism link and radius/base-area meaning
• 12 or below: Relearn the build from V = Bh to V = pi r^2 h before continuing

Common error patterns this quiz is designed to detect

• using diameter as if it were radius
• using circumference instead of base area
• forgetting that the base must be a circle with area pi r^2
• treating height as any visible length instead of perpendicular height
• not understanding that doubling radius multiplies base area by 4, not 2

Next-step routing

If the learner misses mostly:

• Questions 1, 4, 6, or 9: review the derivation and meaning of the formula
• Questions 2, 3, 5, 7, or 10: review formula setup and base-area substitution
• Question 8: review reverse problems and solving for an unknown dimension
• mixed question types across the quiz: use the broader linked volume mastery quizzes for consolidation