Grade 8 Unit Mastery Quiz: Area of the Circular Base

This quiz is a focused mastery check for the subtopic Area of the Circular Base in the cylinder volume derivation sequence.

It is not the full cylinder-volume unit quiz. For broader assessment across deriving and using the cylinder volume formula, see [rea.m08.geometry-measurement.volume.cylinder-volume-derivation-quiz]. For related end-of-unit volume checks, see [rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz] and [rea.m08.geometry-measurement.volume.basic-formulas.prism-bh.unit-mastery-quiz].

Quiz directions

• Grade level: Grade 8 (BC/Ontario-aligned geometry and measurement)
• Time: 25-35 minutes
• Calculator: Optional unless your teacher specifies otherwise
• Unless stated otherwise, give:
  • an exact answer in terms of pi when possible
  • an approximate decimal answer to the nearest tenth when requested
• Total: 20 marks

Learning target

A student demonstrates mastery if they can:

• identify radius or diameter correctly
• use the circular area formula A = pi r^2
• track square units correctly
• move between exact and approximate forms
• solve simple reverse problems involving circular base area
• connect circular base area to cylinder work without needing the full volume formula reteaching

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Questions

1. Direct radius question

A circular base has radius 4 cm. Find its area:

1. in exact form
2. to the nearest tenth

[2 marks]

2. Diameter given

A circular base has diameter 10 m. Find its area in exact form.

[2 marks]

3. Decimal radius

A circular base has radius 2.5 cm. Find its area to the nearest tenth.

[2 marks]

4. Compare two bases

Circle A has radius 3 cm. Circle B has radius 6 cm. How many times as large is the area of Circle B compared with Circle A?

[2 marks]

5. Spot and correct the error

A student says:

"The diameter is 12 cm, so the area is pi(12)^2 = 144pi cm^2."

Explain the mistake and give the correct area.

[2 marks]

6. Reverse problem from exact area

A circular base has area 49pi cm^2. Find its radius.

[2 marks]

7. Reverse problem from decimal area

A circular base has area about 78.5 cm^2. Estimate its radius.

[2 marks]

8. Choose the correct base area for a cylinder

A cylinder has radius 7 cm and height 12 cm. What is the area of one circular base? Do not find the volume.

[2 marks]

9. Real-world interpretation

A round can lid has diameter 8 cm. How much metal is needed to cover just the top of the can, ignoring overlap and thickness? Give the answer to the nearest tenth.

[2 marks]

10. Multi-step readiness question

A cylinder has diameter 14 cm. Its volume can be written as:

V = (area of base) x height

Find the expression for the area of the base, then write the cylinder volume formula for this cylinder in terms of height h. Do not simplify with a decimal approximation.

[2 marks]

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Full marking solutions

1. Direct radius question

Given r = 4 cm

Use A = pi r^2

A = pi(4)^2

A = 16pi cm^2

Approximate value:

A ≈ 16 x 3.14 = 50.24 cm^2

To the nearest tenth: 50.2 cm^2

Answer: 16pi cm^2 and 50.2 cm^2

Marking:

• 1 mark for correct exact form
• 1 mark for correct decimal approximation with units

2. Diameter given

Diameter d = 10 m, so radius r = 5 m

A = pi r^2 = pi(5)^2 = 25pi m^2

Answer: 25pi m^2

Marking:

• 1 mark for finding radius from diameter
• 1 mark for correct area

3. Decimal radius

Given r = 2.5 cm

A = pi r^2 = pi(2.5)^2 = 6.25pi

A ≈ 6.25 x 3.14 = 19.625

To the nearest tenth: 19.6 cm^2

Answer: 19.6 cm^2

Marking:

• 1 mark for correct substitution
• 1 mark for correct approximation and units

4. Compare two bases

Circle A: r = 3 cm

A_A = pi(3)^2 = 9pi

Circle B: r = 6 cm

A_B = pi(6)^2 = 36pi

Compare:

36pi / 9pi = 4

So Circle B has 4 times the area of Circle A.

Answer: 4 times as large

Marking:

• 1 mark for both areas or correct ratio setup
• 1 mark for correct comparison

5. Spot and correct the error

The student used the diameter as if it were the radius.

Given diameter 12 cm, the radius is 6 cm.

Correct area:

A = pi(6)^2 = 36pi cm^2

Answer: The mistake is using diameter instead of radius. Correct area: 36pi cm^2.

Marking:

• 1 mark for identifying the error
• 1 mark for correct corrected area

6. Reverse problem from exact area

Given:

A = 49pi cm^2

Use A = pi r^2

So:

pi r^2 = 49pi

r^2 = 49

r = 7

Radius is positive in this context.

Answer: 7 cm

Marking:

• 1 mark for setting up or cancelling pi correctly
• 1 mark for correct radius

7. Reverse problem from decimal area

Given A ≈ 78.5 cm^2

Use A = pi r^2

r^2 ≈ 78.5 / 3.14 = 25

r ≈ 5

Answer: 5 cm

Marking:

• 1 mark for correct reverse setup
• 1 mark for correct estimate of radius

8. Choose the correct base area for a cylinder

The question asks for one circular base, not the volume.

Given r = 7 cm

A = pi r^2 = pi(7)^2 = 49pi cm^2

Answer: 49pi cm^2

Marking:

• 1 mark for using the base-area formula
• 1 mark for correct result with square units

9. Real-world interpretation

Diameter 8 cm, so radius 4 cm

A = pi r^2 = pi(4)^2 = 16pi

A ≈ 16 x 3.14 = 50.24

To the nearest tenth: 50.2 cm^2

Answer: 50.2 cm^2

Marking:

• 1 mark for converting diameter to radius
• 1 mark for correct decimal area and units

10. Multi-step readiness question

Diameter 14 cm, so radius 7 cm

Area of base:

A = pi(7)^2 = 49pi cm^2

Since V = (area of base) x height,

V = 49pi h

If units are included and h is in cm, then volume is in cm^3.

Answer:

• area of base: 49pi cm^2
• volume formula: V = 49pi h

Marking:

• 1 mark for correct base area
• 1 mark for correct volume expression

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Answer key only

1. 16pi cm^2, 50.2 cm^2
2. 25pi m^2
3. 19.6 cm^2
4. 4 times
5. Used diameter instead of radius; correct area 36pi cm^2
6. 7 cm
7. 5 cm
8. 49pi cm^2
9. 50.2 cm^2
10. 49pi cm^2, V = 49pi h

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Mastery threshold recommendation

Recommended mastery threshold: 17/20 (85%)

Interpretation:

• 17-20: Secure mastery. The student is ready to use circular base area in cylinder derivations and volume problems.
• 14-16: Partial mastery. The student should repair specific gaps, especially radius/diameter confusion, square units, or reverse problems.
• 13 or below: Not yet secure. The student should revisit concept instruction and guided practice before attempting the full cylinder derivation quiz.

What errors mean

Common error patterns to diagnose from this quiz:

• Using diameter directly in pi r^2
• Forgetting to square the radius
• Writing linear units like cm instead of square units like cm^2
• Confusing area of the base with volume of the cylinder
• Struggling to reverse the formula when area is given

Recommended next links

• For the broader end-of-unit cylinder assessment: [rea.m08.geometry-measurement.volume.cylinder-volume-derivation-quiz]
• For volume formula fluency across prisms and cylinders: [rea.m08.geometry-measurement.volume.basic-formulas.unit-mastery-quiz]
• For strengthening the base-area-times-height structure: [rea.m08.geometry-measurement.volume.basic-formulas.prism-bh.unit-mastery-quiz]
• For related cylinder measurement with circular bases in surface area contexts: [rea.m08.geometry-measurement.surface-area.base-perimeter.unit-mastery-quiz]