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.