Mathematics • Year 9 • Unit 4 • Lesson 5
Prisms & Cylinders — Mixed Challenge
Pull together V = (cross-section area) × length for prisms, V = πr²h and SA = 2πr² + 2πrh for cylinders, and the surface-area decomposition idea (nets). You'll spot a mistake in someone else's working and tackle an open-ended challenge about the SAME volume in many different shapes.
1. Mixed problems — choose the right formula
Each question uses a different combination of the prism and cylinder ideas. Decide which formula(s) you need before you start writing. Show your working. 3 marks each
1.1 A cylinder has radius 6 cm and height 9 cm. Find its volume and total surface area. Leave answers in terms of π and also give a decimal to 1 dp.
1.2 A triangular prism has an equilateral-triangle cross-section with side 6 cm (cross-section area = (√3/4) × side² ≈ 15.59 cm²). The prism length is 12 cm. Find its volume to 1 dp.
1.3 A 330 mL drink can holds 330 cm³ of liquid (1 mL = 1 cm³). If the can's diameter is 6.6 cm, find its height to 1 dp (assume it's a cylinder, all 330 mL of liquid fills it).
1.4 A rectangular prism has volume 240 cm³, length 10 cm, and width 6 cm. Find its height. Then find the SA.
1.5 Two cylinders. Cylinder A: r = 4 cm, h = 9 cm. Cylinder B: r = 6 cm, h = 4 cm. Which has the larger volume? By how much (cm³, to 1 dp)?
1.6 A composite solid: a rectangular prism 10 × 5 × 3 cm with a cylindrical hole (radius 1 cm) drilled all the way through, parallel to the 10 cm edge. Find the volume of the solid after the hole is drilled, to 1 dp.
2. Find the mistake
Another student has tried to find the total surface area of a cylinder with radius 5 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 — total SA, r = 5 cm, h = 8 cm:
Line 1: SA = 2π r² + 2π r h
Line 2: SA = π × 5² + 2π × 5 × 8
Line 3: SA = 25π + 80π
Line 4: SA = 105π
Line 5: SA ≈ 330 cm².
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong. (Hint: a cylinder has TWO circular ends, not one — so the "2" in "2π r²" matters.)
(c) Write out the corrected working in full, including the corrected final answer.
Stuck? The formula has 2πr² (TWO ends), but the student wrote πr² (only ONE end). What's the correct first term?3. Open-ended challenge — same volume, different shapes
This question has many valid answers. 4 marks
3.1 Design three different solids that all have a volume of exactly 1000 cm³ = 1 litre. Your three solids must be:
• A cube (find the side length).
• A rectangular (non-cube) prism (choose any two dimensions and find the third).
• A cylinder (choose a radius and find the height, OR choose a height and find the radius).
For each solid you design:
(i) State the dimensions.
(ii) Verify that V = 1000 cm³ (to within 1 cm³).
(iii) Find the total SURFACE AREA of each solid and rank them from smallest SA to largest SA.
Bonus: Which of your three solids uses the LEAST material (smallest SA) for the same volume? Is your answer consistent with the idea that "spheres are the most efficient shape" — and why is the cube/cylinder closer to a sphere than a long thin prism?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cylinder r = 6, h = 9
V = π × 6² × 9 = 324π ≈ 1017.9 cm³.
SA = 2π × 36 + 2π × 6 × 9 = 72π + 108π = 180π ≈ 565.5 cm².
1.2 — Triangular prism (equilateral side 6, length 12)
Cross-section area ≈ 15.59 cm². V = 15.59 × 12 ≈ 187.1 cm³.
1.3 — 330 mL drink can, diameter 6.6 cm
Radius = 3.3 cm. V = π r² h → 330 = π × 3.3² × h → 330 = π × 10.89 × h → h = 330 / (π × 10.89) ≈ 330 / 34.21 ≈ 9.6 cm.
That matches the real height of a standard 330 mL drink can.
1.4 — Prism V = 240, l = 10, w = 6
Height: 240 = 10 × 6 × h → h = 240 / 60 = 4 cm.
SA = 2(10×6 + 10×4 + 6×4) = 2(60 + 40 + 24) = 2(124) = 248 cm².
1.5 — Compare cylinders
Cylinder A: V = π × 16 × 9 = 144π ≈ 452.4 cm³.
Cylinder B: V = π × 36 × 4 = 144π ≈ 452.4 cm³.
They have the SAME volume! Difference = 0 cm³.
Interesting: same πr²h product because 16 × 9 = 36 × 4 = 144.
1.6 — Prism 10 × 5 × 3 with cylindrical hole r = 1, length 10
Prism volume = 10 × 5 × 3 = 150 cm³.
Cylinder (hole) volume = π × 1² × 10 = 10π ≈ 31.42 cm³.
Remaining volume = 150 − 10π ≈ 150 − 31.42 ≈ 118.6 cm³.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) The student forgot the "2" in front of π r² — they wrote π × 5² instead of 2π × 5². A cylinder has TWO circular ends (top AND bottom), so the formula has 2π r², not π r².
(c) Corrected working:
SA = 2π × 5² + 2π × 5 × 8
SA = 50π + 80π
SA = 130π ≈ 408.4 cm².
This is the pitfall flagged in card 3 of the lesson: "forget both circular ends".
3 — Open-ended challenge (sample solutions)
Many valid answers. Here are three solids that all have V = 1000 cm³:
Cube: side = ∛1000 = 10 cm. V = 10³ = 1000 cm³ ✓. SA = 6 × 10² = 600 cm².
Rectangular prism: 20 × 10 × 5 cm. V = 20 × 10 × 5 = 1000 cm³ ✓. SA = 2(20×10 + 20×5 + 10×5) = 2(200 + 100 + 50) = 700 cm².
Cylinder: radius 5 cm, height = 1000 / (π × 25) ≈ 12.73 cm. V = π × 25 × 12.73 ≈ 1000 cm³ ✓. SA = 2π × 5² + 2π × 5 × 12.73 = 50π + 127.3π = 177.3π ≈ 557 cm².
Ranking (smallest SA to largest): Cylinder (~557) < Cube (600) < Rectangular prism (700).
Bonus reasoning: The cylinder with these proportions actually beats the cube here! The "spheres are most efficient" idea means a sphere has the LEAST surface area for a given volume; the further your shape is from a sphere, the more surface area you need. A long thin 20 × 10 × 5 prism is the most "non-sphere" shape, which is why it has the largest SA. Manufacturers use this idea — cylindrical drink cans use less aluminium than a rectangular prism of the same capacity.
Marking: 1 mark per solid (correct V = 1000 cm³ and correct SA); 1 mark for the ranking and reasoning. Up to 4 in total.