Mathematics • Year 8 • Unit 3 • Lesson 9
Surface Area — Mixed Challenge
Six mixed surface-area problems across rectangular prisms, triangular prisms and cylinders. One "find the mistake" and one open-ended packaging design.
1. Mixed problems — choose the right move
Decide whether the shape is a rectangular prism, triangular prism, or cylinder, then pick the matching formula. Show your working. 3 marks each
1.1 A cube has a total surface area of 96 cm². Find the side length of the cube.
1.2 Find the SA of a rectangular prism with l = 15 cm, w = 8 cm, h = 6 cm.
1.3 A cylinder has r = 7 cm and h = 20 cm. Find SA in exact form (nπ) and as a decimal (nearest cm²).
1.4 A triangular prism has a triangular cross-section that is right-angled with legs 5 cm and 12 cm (hypotenuse 13 cm). The prism length is 20 cm. Find SA.
1.5 A cylinder has DIAMETER 14 cm and height 25 cm. Find SA (nearest cm²). (Watch the diameter!)
1.6 A small open-top fish tank (no lid) has l = 60 cm, w = 30 cm, h = 25 cm. Find the area of glass needed for the floor and four walls.
2. Find the mistake
A Year 8 student has tried to find the SA of a cylinder with r = 6 cm and h = 10 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 SA when r = 6 and h = 10:
Line 1: SA = 2πr² + 2πrh
Line 2: SA = 2π(6²) + 2π(6)(10)
Line 3: SA = 2π(36) + 2π(60) = 72π + 120π = 192π
Line 4: SA ≈ 192π ≈ 603 cm³
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected final line, including the corrected units.
Stuck? Revisit lesson § Card 1 — "Units matter". Surface area is always in SQUARE units.3. Open-ended challenge — design two boxes with the same SA
This question has more than one valid answer. 4 marks
3.1 Design two DIFFERENT rectangular boxes that both have a surface area of exactly 100 cm². (The boxes must have different dimensions — for example, you can't list "5 × 5 × 5" twice.)
For each box you design:
(i) State the dimensions l × w × h.
(ii) Show that SA = 2(lw + lh + wh) = 100 cm² exactly.
(iii) Calculate the volume V = lwh.
Bonus: notice that two boxes with the same SA can have very different volumes! Which of your two designs has the larger volume?
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cube SA = 96 cm²
SA = 6s² = 96, so s² = 16, s = 4 cm.
1.2 — Box 15 × 8 × 6
lw = 120, lh = 90, wh = 48. Sum = 258. SA = 2 × 258 = 516 cm².
1.3 — Cylinder r = 7, h = 20
SA = 2π(49) + 2π(7)(20) = 98π + 280π = 378π ≈ 1188 cm².
1.4 — Triangular prism (5-12-13 legs, l = 20)
Triangle area = ½ × 5 × 12 = 30 cm². Two triangles = 60 cm².
Three rectangles = (5 + 12 + 13) × 20 = 30 × 20 = 600 cm².
SA = 60 + 600 = 660 cm².
1.5 — Cylinder, diameter 14, h = 25
Radius r = 14/2 = 7 cm (this is the trap!).
SA = 2π(49) + 2π(7)(25) = 98π + 350π = 448π ≈ 1407 cm².
1.6 — Open-top fish tank 60 × 30 × 25
Floor = 60 × 30 = 1800 cm².
Four walls = 2(60 × 25) + 2(30 × 25) = 3000 + 1500 = 4500 cm².
Total glass = 1800 + 4500 = 6300 cm².
2 — Find the mistake
(a) The mistake is on Line 4.
(b) The student wrote "603 cm³" but surface area is always in SQUARE units (cm²), not cubic units (cm³). Cubic units are for volume.
(c) Corrected final line: SA ≈ 192π ≈ 603 cm². ✓
3 — Design two boxes with SA = 100 cm² (sample solutions)
There are many valid answers. Here are two examples:
Box A: 5 × 5 × 2.5.
Check: 2(25 + 12.5 + 12.5) = 2 × 50 = 100 cm² ✓. V = 5 × 5 × 2.5 = 62.5 cm³.
Box B: 10 × 2 × (50−20)/12 = 10 × 2 × 2.5.
Check: 2(20 + 25 + 5) = 2 × 50 = 100 cm² ✓. V = 10 × 2 × 2.5 = 50 cm³.
Comparison: Box A holds 62.5 cm³ but Box B holds only 50 cm³, even though both have the same SA. Box A is more efficient — a more cube-like shape holds more volume per surface area than a long, thin shape.
Marking: 1 mark per valid box with correct SA check (up to 2 marks). 1 mark for both volumes calculated. 1 mark for comparing which is more efficient.