Mathematics • Year 8 • Unit 3 • Lesson 11
Cylinders in the Real World
Use V = πr²h where cylinders actually appear: water tanks, soup cans, swimming pools and rainwater barrels. Then explain your thinking in your own words.
1. Word problems
Each problem hides a cylinder. Sketch and label r and h, then apply V = πr²h. Remember: 1 cm³ = 1 mL, 1 m³ = 1000 L. Show working — a single final answer with no working earns only half marks.
1.1 — Water tank. A cylindrical rainwater tank has radius 1.5 m and height 2 m.
(a) Find the tank's volume in m³ to 2 decimal places.
(b) Convert your answer to litres (1 m³ = 1000 L).
(c) If a family uses 300 L per day, how many full days does the tank last? 3 marks
1.2 — Soup can. A standard soup can has diameter 7.4 cm and height 11 cm.
(a) Find the radius first.
(b) Find the volume of soup the can holds to 1 decimal place.
(c) Convert to mL (1 cm³ = 1 mL). Does this match the "420 mL" label on the can? 3 marks
1.3 — Swimming pool. A circular kids' pool has diameter 3 m and depth 0.4 m.
(a) Find the radius.
(b) Find the volume of water it holds (when full) in m³ to 2 decimal places.
(c) Convert to litres. If a tap fills at 20 L per minute, how long does it take to fill? 3 marks
1.4 — Drink bottle. A cylindrical sports drink bottle has radius 3.5 cm and height 22 cm.
(a) Find its volume in cm³ to 1 decimal place.
(b) Convert to mL.
(c) Does it hold more or less than 1 litre? By how much? 3 marks
1.5 — Find the height. A cylindrical paint can must hold exactly 4 litres (= 4000 cm³). The base of the can has radius 8 cm.
(a) Rearrange V = πr²h to make h the subject.
(b) Find the required height of the can to 1 decimal place. 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate has been asked to find the volume of a cylinder with diameter 10 cm and height 6 cm. They write "V = π × 10² × 6 = 600π ≈ 1885 cm³." In your own words, explain (i) what mistake they have made, (ii) what the correct volume is and how to get it, and (iii) why their wrong answer is exactly four times too large. Use the phrase "halve the diameter first" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Water tank
(a) V = π × 1.5² × 2 = 4.5π ≈ 14.14 m³.
(b) 14.14 × 1000 = 14,137 L (more precisely 14,137.2 L).
(c) 14,137 ÷ 300 = 47.12, so the tank lasts 47 full days.
1.2 — Soup can
(a) r = 7.4 ÷ 2 = 3.7 cm.
(b) V = π × 3.7² × 11 = π × 13.69 × 11 = 150.59π ≈ 473.1 cm³.
(c) 473.1 cm³ = 473.1 mL. This is more than 420 mL because manufacturers leave headspace at the top of the can for safe sealing and to allow for liquid expansion.
1.3 — Swimming pool
(a) r = 3 ÷ 2 = 1.5 m.
(b) V = π × 1.5² × 0.4 = π × 2.25 × 0.4 = 0.9π ≈ 2.83 m³.
(c) 2.83 × 1000 = 2827 L. Time = 2827 ÷ 20 ≈ 141 minutes (about 2 hours 21 minutes).
1.4 — Drink bottle
(a) V = π × 3.5² × 22 = π × 12.25 × 22 = 269.5π ≈ 846.7 cm³.
(b) = 846.7 mL.
(c) Less than 1 L. Short by 1000 − 846.7 = 153.3 mL.
1.5 — Paint can height
(a) Starting from V = πr²h, divide both sides by πr²: h = V ÷ (πr²).
(b) h = 4000 ÷ (π × 8²) = 4000 ÷ (π × 64) = 4000 ÷ 201.06 ≈ 19.9 cm.
2.1 — Explain your thinking (sample response)
The classmate used the diameter (10 cm) instead of the radius. The formula V = πr²h needs the radius, so they must halve the diameter first: r = 10 ÷ 2 = 5 cm. The correct calculation is V = π × 5² × 6 = π × 25 × 6 = 150π ≈ 471.2 cm³. Their answer was exactly four times too large because using 10 instead of 5 makes r² become 100 instead of 25 — and 100 ÷ 25 = 4. Whenever the radius is doubled (or halved) by mistake, the volume changes by a factor of 4, because the radius is squared in the formula.
Marking: 1 mark for identifying "used d instead of r"; 1 mark for showing r = 5 and V ≈ 471.2 cm³; 1 mark for the "factor of 4" explanation (because r is squared); 1 mark for using the phrase "halve the diameter first" in a clear sentence.