Mathematics • Year 8 • Unit 3 • Lesson 11
Cylinders — Mixed Challenge
Pull everything from Lesson 11 together: volume from r and h, from d and h, missing dimensions, and capacity conversions. Six mixed problems, one "find the mistake", and one open-ended design challenge.
1. Mixed problems — choose the right move
Each question pulls a different idea from Lesson 11. Decide which approach applies before you start writing. Show your working. Use π ≈ 3.14159. 3 marks each
1.1 A cylinder has r = 7 cm and h = 12 cm. Find its volume to 1 decimal place.
1.2 A cylinder has diameter 16 cm and height 9 cm. Find its volume to 1 decimal place.
1.3 A cylinder has volume 942.5 cm³ and radius 5 cm. Find its height to the nearest whole number.
1.4 A cylindrical water bottle holds 750 mL. Its base has radius 4 cm. Find its height to 1 decimal place. (Hint: 750 mL = 750 cm³.)
1.5 A cylindrical fish tank has r = 20 cm and h = 35 cm. Find its capacity in litres (1 L = 1000 cm³) to 1 decimal place.
1.6 A small cylinder has r = 3 cm and h = 5 cm. A second cylinder has the same height but double the radius (r = 6 cm). Without a calculator, predict the ratio of their volumes, then check by calculating both.
2. Find the mistake
A Year 8 student has tried to find the volume of a cylinder with diameter 12 cm and height 8 cm. Their working is shown below. Exactly one line contains a mistake. Spot it, explain why it's wrong, then re-do the working correctly. 3 marks
Student's working — find V when d = 12 cm and h = 8 cm:
Line 1: V = πr²h
Line 2: V = π × 12² × 8
Line 3: V = π × 144 × 8 = 1152π
Line 4: V ≈ 3619.1 cm³
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? Revisit lesson § Card 9 — "Using diameter instead of radius" is the most common pitfall. The student forgot to halve d before squaring.3. Open-ended challenge — design a 1 L cylinder
This question has more than one valid answer. 4 marks
3.1 Design a cylindrical drink container that holds exactly 1 litre (= 1000 cm³). Find three different combinations of radius and height that work.
For each design:
(i) State r and h (in cm, with at least one decimal place).
(ii) Show the check: V = πr²h ≈ 1000 cm³ (within ±5 cm³).
(iii) Describe what the cylinder would look like — tall and narrow, short and wide, or roughly square (h ≈ d)?
Bonus: at least one of your designs must have h between 18 cm and 25 cm (a typical drink-bottle shape).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — r = 7, h = 12
V = π × 49 × 12 = 588π ≈ 1847.3 cm³.
1.2 — d = 16, h = 9
r = 8 cm. V = π × 64 × 9 = 576π ≈ 1809.6 cm³.
1.3 — V = 942.5, r = 5
h = V ÷ (πr²) = 942.5 ÷ (π × 25) = 942.5 ÷ 78.54 ≈ 12 cm.
1.4 — V = 750 cm³, r = 4
h = 750 ÷ (π × 16) = 750 ÷ 50.27 ≈ 14.9 cm.
1.5 — Fish tank r = 20, h = 35
V = π × 400 × 35 = 14,000π ≈ 43,982.3 cm³. Capacity = 43,982.3 ÷ 1000 ≈ 44.0 L.
1.6 — Doubling the radius
Prediction: doubling r quadruples V (because r is squared). Ratio = 1 : 4.
Small: V₁ = π × 9 × 5 = 45π ≈ 141.4 cm³.
Large: V₂ = π × 36 × 5 = 180π ≈ 565.5 cm³.
Check: 180 ÷ 45 = 4 ✓.
2 — Find the mistake
(a) The mistake is on Line 2 (and carries into Lines 3 and 4).
(b) The student used the diameter (12 cm) instead of the radius. The formula needs r, so they should have halved d first: r = 12 ÷ 2 = 6 cm. Using d = 12 instead of r = 6 makes r² four times too big (144 instead of 36).
(c) Corrected working:
V = πr²h
r = 12 ÷ 2 = 6 cm
V = π × 6² × 8 = π × 36 × 8 = 288π
V ≈ 904.8 cm³. ✓
Sanity check: the wrong answer (3619.1) is exactly 4 × the correct (904.8) — that confirms the diameter-vs-radius error.
3 — 1 L cylinder design (sample solution)
Many valid designs exist. Three good examples:
Design 1 — tall and narrow: r = 4 cm, h = 19.9 cm.
Check: V = π × 16 × 19.9 = π × 318.4 ≈ 1000.1 cm³ ✓ (drink-bottle shape).
Design 2 — short and wide: r = 6 cm, h = 8.8 cm.
Check: V = π × 36 × 8.8 = π × 316.8 ≈ 995.4 cm³ ✓ (paint-can shape).
Design 3 — roughly square: r = 5 cm, h = 12.7 cm.
Check: V = π × 25 × 12.7 = π × 317.5 ≈ 997.5 cm³ ✓ (tin-can shape, h ≈ d = 10 cm? — actually h ≈ 12.7 > d).
Marking: 1 mark per valid design with correct check (volume within ±5 cm³ of 1000) up to 3 marks. 1 bonus mark for including at least one design with h between 18 cm and 25 cm (typical drink bottle).