Mathematics • Year 10 • Unit 1 • Lesson 18
Volume of Prisms and Cylinders — Mixed Challenge
Pull together every idea from Lesson 18: the universal V = Abase × h, special cases lwh, ½bh·L and πr²h, unit conversion across cm³/m³/mL/L/kL, and choosing the cheaper container. Choose the right tool for each problem, spot another student's unit-conversion blunder, then design two containers that hold the same litres.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 18. Decide which formula or conversion applies before you start writing. 3 marks each
1.1 A rectangular prism has dimensions 25 cm by 18 cm by 12 cm. Find its volume in (a) cm³, (b) litres.
1.2 A cylinder has radius 7 cm and height 20 cm. Find its volume in cm³ (exact in π and to 1 d.p.).
1.3 A triangular prism has a right-angled base triangle with legs 9 cm and 12 cm. The prism is 16 cm long. Find its volume in cm³.
1.4 Convert: (a) 7500 cm³ to L, (b) 2.4 m³ to kL, (c) 0.65 L to cm³.
1.5 A cylindrical can of paint has diameter 16 cm and holds 4 L of paint. What is the height of the paint inside the can (assume full)? Give the height in cm to 1 d.p.
1.6 A backyard concrete slab measures 6 m by 4 m and is 100 mm thick. Concrete is sold by the cubic metre. How many m³ of concrete must be ordered?
2. Find the mistake
Another Year 10 student has tried to find the capacity in litres of a rectangular fish tank measuring 100 cm × 40 cm × 50 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 the capacity in litres of a 100 × 40 × 50 cm tank:
Line 1: V = 100 × 40 × 50 = 200 000 cm³
Line 2: 200 000 cm³ = 200 000 mL
Line 3: Capacity = 200 000 ÷ 100 = 2000 L
(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? 1 L = 1000 mL — divide by 1000, not 100.3. Open-ended challenge — design two 5-litre containers
This question has many valid answers. Be creative but show every number. 4 marks
3.1 A kitchenware company wants two new 5-litre containers. One must be a rectangular prism and the other must be a cylinder. Both must have a capacity of exactly 5 L (= 5000 cm³, within ±50 cm³).
For each container:
(a) Choose realistic dimensions (rectangular: state l, w, h in cm; cylinder: state r and h in cm).
(b) Calculate the volume in cm³ and confirm it is within ±50 cm³ of 5000.
(c) Convert your volume to litres.
(d) State one practical advantage of your chosen shape (e.g. fits in a fridge door, stacks easily, no corners to clean).
Constraints: no dimension may exceed 40 cm; rectangular prism must have at least one dimension ≥ 10 cm; cylinder radius must be between 5 cm and 15 cm.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Rectangular prism 25 × 18 × 12
(a) V = 25 × 18 × 12 = 5400 cm³.
(b) 5400 cm³ = 5400 mL = 5.4 L.
1.2 — Cylinder (r = 7, h = 20)
V = π × 49 × 20 = 980π cm³ ≈ 3078.8 cm³ (1 d.p.).
1.3 — Triangular prism (legs 9, 12; length 16)
Abase = ½ × 9 × 12 = 54 cm². V = 54 × 16 = 864 cm³.
1.4 — Conversions
(a) 7500 cm³ ÷ 1000 = 7.5 L.
(b) 2.4 m³ = 2.4 kL (1 m³ = 1 kL).
(c) 0.65 L × 1000 = 650 mL = 650 cm³.
1.5 — Paint can (d = 16 cm, V = 4 L)
r = 8 cm. V = 4 L = 4000 cm³.
h = V ÷ (πr²) = 4000 ÷ (64π) ≈ 19.9 cm (1 d.p.).
1.6 — Concrete slab 6 m × 4 m × 100 mm
100 mm = 0.1 m. V = 6 × 4 × 0.1 = 2.4 m³ of concrete.
2 — Find the mistake
(a) The mistake is on Line 3.
(b) 1 L = 1000 mL (not 100 mL). To convert mL → L the student should divide by 1000, not by 100.
(c) Corrected working:
V = 100 × 40 × 50 = 200 000 cm³ ✓
200 000 cm³ = 200 000 mL ✓
Capacity = 200 000 ÷ 1000 = 200 L.
Quick sanity check: a tank 1 m × 0.4 m × 0.5 m = 0.2 m³ = 200 L — exactly what the corrected working gives.
3 — Open-ended challenge (sample solutions)
We need a rectangular prism container and a cylinder, both holding ≈ 5 L.
Container A — rectangular prism 25 × 20 × 10 cm
(a) l = 25, w = 20, h = 10.
(b) V = 25 × 20 × 10 = 5000 cm³ ✓.
(c) = 5 L.
(d) Stacks neatly in cupboards; flat sides make it easy to label.
Container B — cylinder r = 10 cm, h ≈ 15.9 cm
(a) r = 10 cm, h = 15.9 cm.
(b) V = π(10)²(15.9) = 1590π ≈ 4995 cm³ (within ±50 of 5000) ✓.
(c) ≈ 5.0 L.
(d) No corners — easier to clean and pour; fits common drink-bottle holders.
Marking: 1 for rectangular prism dimensions and correct volume; 1 for cylinder dimensions and correct volume; 1 for valid litre conversion on each; 1 for a sensible practical advantage statement. All constraints respected. Other valid dimension sets accepted.