Layering a Cylinder: Grade 8 Unit Mastery Quiz with Full Solutions

Position in the unit

This quiz is a focused mastery check for Layering a Cylinder. It should be used after the learner has studied the linked unit overview [rea.m08.geometry-measurement.volume.cylinder-derivation.layering] and before or alongside the broader derivation and formula quizzes.

For direct instruction on the concept, use:

• [rea.m08.geometry-measurement.volume.cylinder-derivation.layering]

For broader assessment beyond the layering focus, use:

• [rea.m08.geometry-measurement.volume.cylinder-volume-derivation-quiz]
• [rea.m08.geometry-measurement.volume.basic-formulas.cylinders.unit-mastery-quiz]
• [rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz]

Quiz directions

• Grade level: Grade 8 (Canadian curriculum)
• Show reasoning, not only final answers.
• Unless stated otherwise, use exact answers in terms of pi, then give a decimal approximation if helpful.
• Total suggested value: 24 marks

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Questions

1. Explain the layering idea. (2 marks)

A cylinder has radius r and height h. Explain in words why its volume can be found by multiplying the area of one circular layer by the number of layers.

2. Identify the volume expression from layers. (2 marks)

A can is made of 12 identical circular layers, each layer 3 cm thick. The area of each circular layer is 25pi cm^2.

1. What is the total height of the can?
2. What is its volume?

3. Match the layer model to the formula. (2 marks)

A cylinder has radius 4 cm and height 9 cm.

1. What is the area of one circular layer?
2. Using the layering idea, find the volume.

4. Spot and repair the error. (2 marks)

A student says: "A cylinder is made of circular layers, so its volume is 2pi r h."

Explain what the student mixed up, and write the correct volume formula with a brief reason.

5. Diameter versus radius in a layer model. (3 marks)

A cylinder has diameter 10 m and height 7 m.

1. Find the area of one circular layer.
2. Find the volume.
3. State the units for the volume.

6. Compare two cylinders by layering. (3 marks)

Cylinder A and Cylinder B have the same height, 8 cm.

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

1. Find the volume of each cylinder.
2. How many times larger is the volume of Cylinder B than Cylinder A?
3. Explain the result using the layer idea.

7. Reverse problem from volume and height. (3 marks)

A cylinder has volume 180pi cm^3 and height 5 cm.

1. Find the area of one circular layer.
2. Find the radius.

8. Fractional layers and unit sense. (2 marks)

A cylindrical container has base area 16pi cm^2 and height 2.5 cm.

Use the layer model to find its volume and explain why a non-whole-number height still works.

9. Real-world application. (3 marks)

A round storage tube has inside radius 7 cm and inside height 30 cm.

1. Find the inside volume in exact form.
2. Find the inside volume to the nearest tenth of a cubic centimetre.
3. Explain what each factor in your calculation represents physically.

10. Write a complete derivation in one paragraph. (2 marks)

Write a short paragraph showing how the formula V = pi r^2 h comes from viewing a cylinder as stacked circular layers. Your answer must mention base area, height, and cubic units.

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Full marking solutions

1. Explain the layering idea. (2 marks)

Solution: A cylinder can be seen as many equal circular slices stacked from bottom to top. Each slice has the same area as the circular base, which is pi r^2. Multiplying the area of one layer by the total height gives the total space filled, so the volume is V = pi r^2 h.

Marking:

• 1 mark for identifying equal circular layers with area pi r^2
• 1 mark for connecting multiplication by height to total volume

2. Identify the volume expression from layers. (2 marks)

Solution:

1. Total height: 12 x 3 = 36 cm
2. Volume: 25pi x 36 = 900pi cm^3

Answer: 36 cm, 900pi cm^3

Marking:

• 1 mark for correct height
• 1 mark for correct volume with units

3. Match the layer model to the formula. (2 marks)

Solution:

1. Area of one circular layer: pi r^2 = pi(4)^2 = 16pi cm^2
2. Volume: 16pi x 9 = 144pi cm^3

Answer: 16pi cm^2, 144pi cm^3

Marking:

• 1 mark for layer area
• 1 mark for volume

4. Spot and repair the error. (2 marks)

Solution: The student mixed up circumference with area. The expression 2pi r is the circumference of a circle, not the area of a circular layer. Volume needs base area x height, not circumference x height. The correct formula is V = pi r^2 h because each layer is a circle with area pi r^2, and the cylinder is built by stacking those layers through height h.

Marking:

• 1 mark for identifying the confusion with circumference
• 1 mark for correct repaired formula and reason

5. Diameter versus radius in a layer model. (3 marks)

Solution: Diameter = 10 m, so radius r = 5 m.

1. Area of one layer: pi r^2 = pi(5)^2 = 25pi m^2
2. Volume: 25pi x 7 = 175pi m^3
3. Units: cubic metres, m^3

Answer: 25pi m^2, 175pi m^3, units m^3

Marking:

• 1 mark for using radius 5 m
• 1 mark for correct layer area
• 1 mark for correct volume and cubic units

6. Compare two cylinders by layering. (3 marks)

Solution: Cylinder A: V_A = pi(3)^2(8) = pi(9)(8) = 72pi cm^3

Cylinder B: V_B = pi(6)^2(8) = pi(36)(8) = 288pi cm^3

2. Comparison: 288pi / 72pi = 4 So Cylinder B has 4 times the volume of Cylinder A.

3. Explanation: Both cylinders have the same height, so they have the same number of layers. Cylinder B's circular layers have radius twice as large, so each layer has area four times as large because area depends on r^2. Therefore the total volume is four times as large.

Marking:

• 1 mark for both correct volumes
• 1 mark for correct factor of 4
• 1 mark for explanation using equal height and larger layer area

7. Reverse problem from volume and height. (3 marks)

Solution: Given V = 180pi cm^3 and h = 5 cm.

1. Area of one layer: base area = volume / height = 180pi / 5 = 36pi cm^2

2. Find radius: pi r^2 = 36pi r^2 = 36 r = 6 cm

Answer: layer area 36pi cm^2, radius 6 cm

Marking:

• 1 mark for dividing volume by height correctly
• 1 mark for equation pi r^2 = 36pi
• 1 mark for radius 6 cm

8. Fractional layers and unit sense. (2 marks)

Solution: Volume = base area x height = 16pi x 2.5 = 40pi cm^3

A non-whole-number height still works because volume scales continuously with height. The cylinder does not need to be imagined as only whole-number layers; it can be made of very thin equal layers, so multiplying base area by 2.5 cm is still valid.

Answer: 40pi cm^3

Marking:

• 1 mark for correct volume
• 1 mark for correct explanation of why fractional height is valid

9. Real-world application. (3 marks)

Solution: Radius r = 7 cm, height h = 30 cm

1. Exact volume: V = pi r^2 h = pi(7)^2(30) = pi(49)(30) = 1470pi cm^3

2. Decimal approximation: 1470pi approx 4618.1 cm^3

3. Physical meaning:

• pi r^2 is the area of the circular opening or base of the tube
• h is how far that same cross-section extends through the tube
• Their product gives the amount of three-dimensional space inside the tube

Answer: 1470pi cm^3, approximately 4618.1 cm^3

Marking:

• 1 mark for exact form
• 1 mark for correct decimal approximation
• 1 mark for physical interpretation of factors

10. Write a complete derivation in one paragraph. (2 marks)

Sample solution: A cylinder can be thought of as a stack of identical circular layers from bottom to top. Each layer has the same area as the base, and the base area is pi r^2. Stacking that equal cross-section through a height of h means the total volume is pi r^2 x h, so V = pi r^2 h. Since area is measured in square units and height is measured in linear units, the result is in cubic units.

Marking:

• 1 mark for correct conceptual derivation using layers, base area, and height
• 1 mark for mentioning cubic units

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Scoring and mastery recommendation

Total: 24 marks

Recommended mastery threshold

Recommend 20/24 or higher, with no major conceptual error on Questions 1, 4, 7, or 10.

Interpretation

• 22-24: Strong mastery. Learner can explain and use the layering derivation confidently.
• 20-21: Secure mastery. Small arithmetic or wording issues only.
• 16-19: Partial mastery. Revisit the layer model, especially the link base area x height.
• 15 or below: Not yet mastered. Return first to [rea.m08.geometry-measurement.volume.cylinder-derivation.layering], then reattempt this quiz.

Common error patterns this quiz is designed to detect

• Using diameter as if it were radius
• Confusing circle area pi r^2 with circumference 2pi r
• Treating volume units as square units instead of cubic units
• Using the formula without being able to explain the layered meaning
• Struggling to reverse from volume back to base area or radius

Suggested use in the Rea sequence

Use this record as the topic-specific assessment for Layering a Cylinder. For broader mixed review across cylinder volume derivation and formula use, assign the linked records rather than extending this quiz with duplicate items.