Units and Reasonableness Checks for Cylinder Volume: Extended Worked Examples

Grade 8 focus

This record is for Grade 8 learners studying cylinder volume in the Canadian curriculum context. It does not re-teach the full derivation of V = pi r^2 h; use the linked lesson and unit overview for that foundation. Here, the goal is narrower and practical: when you calculate cylinder volume, make sure the units are correct and the answer is sensible.

Use this record with the linked material

Build on these records rather than replacing them:

• rea.m08.geometry-measurement.volume.cylinder-derivation.units-checks.lesson: core lesson on unit tracking and reasonableness checks for cylinders.
• rea.m08.geometry-measurement.volume.cylinder-derivation.units-checks: unit overview for this topic.
• rea.m08.geometry-measurement.volume.basic-formulas.worked-examples: broader worked examples for prism and cylinder formulas.
• rea.m08.geometry-measurement.volume.basic-formulas.units-reasonableness.worked-examples: additional unit-and-sense-check practice across volume problems.

Quick procedure: the check-before-you-accept routine

For every cylinder-volume problem:

1. Identify r and h carefully.
2. Make all length units match before substituting.
3. Compute V = pi r^2 h.
4. Attach cubic units such as cm^3, m^3, or in^3.
5. Do a reasonableness check:
   • Is the answer positive?
   • Should it be more or less than a nearby easy estimate?
   • If one dimension is bigger, should the volume be bigger?
   • Are the units cubic, not square or linear?

Worked examples

Example 1: Direct substitution with whole numbers

Problem. A cylinder has radius 3 cm and height 8 cm. Find its volume.

Solution.

Use V = pi r^2 h.

Substitute r = 3 cm and h = 8 cm:

V = pi(3 cm)^2(8 cm)

V = pi(9 cm^2)(8 cm)

V = 72pi cm^3

Approximate:

V approx 72 x 3.14 = 226.08 cm^3

Answer: 72pi cm^3 or about 226.1 cm^3.

Reasoning commentary. Squaring the radius turns cm into cm^2; multiplying by height gives cm^3. That is exactly what we want for volume.

Reasonableness check. Since pi is a little more than 3, 72pi should be a little more than 216. The result 226.08 fits.

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Example 2: Decimal dimensions

Problem. A can has radius 4.5 cm and height 10 cm. Find its volume.

Solution.

V = pi r^2 h

V = pi(4.5 cm)^2(10 cm)

V = pi(20.25 cm^2)(10 cm)

V = 202.5pi cm^3

Approximate:

V approx 202.5 x 3.14 = 635.85 cm^3

Answer: 202.5pi cm^3 or about 635.9 cm^3.

Reasoning commentary. It is worth squaring 4.5 carefully. A common slip is to double it or square only the 4. Write the square explicitly before multiplying by height.

Reasonableness check. Since 4.5 is close to 5, a quick estimate is pi x 25 x 10 = 250pi approx 785. The exact answer should be somewhat smaller because 4.5^2 = 20.25, not 25. 635.9 is reasonable.

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Example 3: Diameter is given, not radius

Problem. A cylinder has diameter 12 m and height 7 m. Find its volume.

Solution.

The formula uses radius, so first convert diameter to radius:

r = 12/2 = 6 m

Now use the formula:

V = pi(6 m)^2(7 m)

V = pi(36 m^2)(7 m)

V = 252pi m^3

Approximate:

V approx 252 x 3.14 = 791.28 m^3

Answer: 252pi m^3 or about 791.3 m^3.

Reasoning commentary. The important choice is to pause and convert diameter to radius before substituting. If 12 is used directly as the radius, the answer becomes four times too large because area depends on r^2.

Reasonableness check. Base area is 36pi, which is a bit more than 108. Multiplying by height 7 gives a bit more than 756. The result 791.28 is sensible.

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Example 4: Mixed units must be fixed first

Problem. A cylinder has radius 20 cm and height 1.5 m. Find its volume in cm^3.

Solution.

The units do not match. Convert height to centimetres:

1.5 m = 150 cm

Now calculate:

V = pi(20 cm)^2(150 cm)

V = pi(400 cm^2)(150 cm)

V = 60000pi cm^3

Approximate:

V approx 60000 x 3.14 = 188400 cm^3

Answer: 60000pi cm^3 or about 188400 cm^3.

Reasoning commentary. Volume formulas only work cleanly when all lengths are in the same unit. Converting after the calculation is possible, but converting first reduces mistakes.

Reasonableness check. A radius of 20 cm means a diameter of 40 cm, and a height of 150 cm is much taller than it is wide. A volume around 188000 cm^3 is plausible for a large container.

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Example 5: Same problem, answer in m^3

Problem. A cylinder has radius 20 cm and height 1.5 m. Find its volume in m^3.

Solution.

Convert the radius to metres:

20 cm = 0.2 m

Now use metres throughout:

V = pi(0.2 m)^2(1.5 m)

V = pi(0.04 m^2)(1.5 m)

V = 0.06pi m^3

Approximate:

V approx 0.06 x 3.14 = 0.1884 m^3

Answer: 0.06pi m^3 or about 0.1884 m^3.

Reasoning commentary. This example shows the same solid as Example 4. The number changed a lot because the unit changed from cm^3 to m^3, not because the solid changed.

Reasonableness check. Since 1 m^3 = 1,000,000 cm^3, the answer in cubic metres should be much smaller numerically than the answer in cubic centimetres. That is exactly what happens.

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Example 6: Decide which answer is reasonable

Problem. A student calculates the volume of a cylinder with radius 5 cm and height 9 cm and gets one of these answers:

• A. 225pi cm^2
• B. 225pi cm^3
• C. 90pi cm^3

Which answer is correct, and why are the others unreasonable?

Solution.

Compute the volume:

V = pi r^2 h = pi(5 cm)^2(9 cm)

V = pi(25 cm^2)(9 cm)

V = 225pi cm^3

So B is correct.

Why A is unreasonable:

• It has cm^2, which is an area unit.
• Volume must have cubic units.

Why C is unreasonable:

• 90pi would come from multiplying incorrectly.
• Since 5^2 = 25, not 10, the factor before pi should be 25 x 9 = 225.

Answer: 225pi cm^3.

Reasoning commentary. A correct number with wrong units is still wrong. Unit checking is part of the mathematics, not just a label at the end.

Reasonableness check. The base area is about 78.5 cm^2. Multiplying by height 9 gives about 706.5 cm^3, and 225pi approx 706.5, so the answer fits.

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Example 7: Back-checking an obviously impossible answer

Problem. A cylinder has radius 2 cm and height 4 cm. A student says the volume is 160pi cm^3. Explain why this is not reasonable, then find the correct volume.

Solution.

First compute correctly:

V = pi(2 cm)^2(4 cm)

V = pi(4 cm^2)(4 cm)

V = 16pi cm^3

Approximate:

V approx 50.24 cm^3

Now explain why 160pi cm^3 is unreasonable.

• The cylinder is small: radius 2 cm, height 4 cm.
• The base area is only 4pi, about 12.56 cm^2.
• Multiplying by height 4 should give about 50 cm^3, not about 503 cm^3.

Answer: The correct volume is 16pi cm^3 or about 50.2 cm^3.

Reasoning commentary. Estimation catches place-value mistakes very well. Before accepting a large result, compare it with the size of the actual object.

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Example 8: Comparing two cylinders without full calculation first

Problem. Cylinder A has radius 4 cm and height 10 cm. Cylinder B has radius 8 cm and height 10 cm. Which has greater volume, and by what factor?

Solution.

Because the heights are the same, compare r^2.

For Cylinder A:

r^2 = 4^2 = 16

For Cylinder B:

r^2 = 8^2 = 64

Now compare:

64/16 = 4

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

If we calculate fully:

V_A = pi(4^2)(10) = 160pi cm^3

V_B = pi(8^2)(10) = 640pi cm^3

And indeed:

640pi / 160pi = 4

Answer: Cylinder B has the greater volume, and it is 4 times as large.

Reasoning commentary. Doubling the radius does not double the volume when height stays fixed; it multiplies the base area by 2^2 = 4.

Reasonableness check. This is a structure check, not just an arithmetic check. Since radius is squared, changes in radius have a stronger effect than many students expect.

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Example 9: Solving from volume and checking units carefully

Problem. A cylinder has volume 314 cm^3 and height 10 cm. Using pi approx 3.14, find the radius.

Solution.

Start with:

V = pi r^2 h

Substitute known values:

314 = 3.14 x r^2 x 10

314 = 31.4r^2

Divide both sides by 31.4:

r^2 = 10

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

r = sqrt(10) approx 3.16

Answer: The radius is about 3.16 cm.

Reasoning commentary. When solving backwards, keep the physical meaning in mind. Even if an algebra step could produce -3.16, geometry rules it out.

Reasonableness check. A radius a little bigger than 3 cm gives base area a little bigger than 28 cm^2; multiplying by height 10 cm gives a volume a little bigger than 280 cm^3. That matches 314 cm^3.

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Example 10: Challenge problem with comparison to a bounding box

Problem. A cylinder fits exactly inside a rectangular prism with length 12 cm, width 12 cm, and height 15 cm. Find the cylinder's volume and use the prism to check whether the answer is reasonable.

Solution.

A cylinder that fits exactly inside a 12 cm by 12 cm square base has:

• diameter 12 cm
• radius 6 cm
• height 15 cm

Now calculate the cylinder volume:

V = pi(6 cm)^2(15 cm)

V = pi(36 cm^2)(15 cm)

V = 540pi cm^3

Approximate:

V approx 540 x 3.14 = 1695.6 cm^3

Now compare with the surrounding prism:

V_prism = 12 x 12 x 15 = 2160 cm^3

Since the cylinder is inside the prism, its volume must be less than 2160 cm^3.

And it is:

1695.6 < 2160

Answer: 540pi cm^3 or about 1695.6 cm^3.

Reasoning commentary. This is a stronger reasonableness check. Instead of only estimating, compare with a shape you know must be larger.

Reasonableness check. If someone got an answer bigger than 2160 cm^3, that would be impossible.

Common misconceptions and repairs

1. Using diameter in place of radius

Error. Substituting the diameter directly into pi r^2 h.

Repair. Circle the given measure and ask: "Is this the distance across or from centre to edge?" If it is across, divide by 2 first.

2. Writing square units for volume

Error. Ending with cm^2 or m^2.

Repair. Track units during the calculation: r^2 gives square units, then multiplying by height gives cubic units.

3. Mixing units

Error. Using r in centimetres and h in metres in the same substitution.

Repair. Convert all lengths before substituting. Write a note: "same units first."

4. Expecting volume to change linearly with radius

Error. Thinking that doubling radius doubles volume.

Repair. Highlight the exponent: r^2. If radius doubles and height stays the same, volume is multiplied by 4.

5. Accepting impossible answers

Error. Keeping a very large or very small answer without checking.

Repair. Always estimate or compare to a simple benchmark, such as a nearby prism or an easy rounded calculation.

Short practice with answers

1. A cylinder has radius 2 cm and height 9 cm. Find the volume.

Answer: 36pi cm^3 approx 113.0 cm^3

2. A cylinder has diameter 14 m and height 5 m. Find the volume.

Answer: 245pi m^3 approx 769.3 m^3

3. A cylinder has radius 30 cm and height 0.8 m. Find the volume in cm^3.

Answer: 72000pi cm^3 approx 226080 cm^3

4. A student writes V = pi(6)^2(4) = 144pi cm^2. What must be corrected?

Answer: The units must be cm^3, not cm^2.

5. Two cylinders have the same height. One has radius 3 cm; the other has radius 6 cm. How many times larger is the second volume?

Answer: 4 times as large.

Takeaway

For Grade 8 cylinder volume problems, the calculation is only half the job. A complete solution includes correct substitution, consistent units, cubic units in the answer, and a reasonableness check that could catch a setup mistake before the answer is accepted.