Mathematics • Year 10 • Unit 1 • Lesson 16
Surface Area of Prisms — Mixed Challenge
Pull together every idea from Lesson 16: SA = 2(lw + lh + wh), triangular prism nets, open prisms with missing faces, paint/cardboard cost problems and unit conversion (cm² ↔ m²). Choose the right tool for each problem, spot another student's mistake, then design your own minimum-material packaging.
1. Mixed problems — choose the right tool
Each question uses a different idea from Lesson 16. Decide which formula or method applies before you start writing. Show your working. 3 marks each
1.1 A closed rectangular box has dimensions 14 cm by 9 cm by 6 cm. Calculate the surface area.
1.2 A cube has side length 9 cm. Calculate its total surface area.
1.3 An open-top wooden crate measures 60 cm by 40 cm by 30 cm. Find the area of wood needed.
1.4 A triangular prism has a base triangle with base 10 cm and perpendicular height 6 cm (third side 11.66 cm). The prism is 18 cm long. Find the total surface area.
1.5 A closed display case is 1.5 m by 0.6 m by 0.4 m. Glass costs $95 per m². Find the cost of the glass for the case.
1.6 A painter quotes $7 per m² to paint an external shed wall and roof. The shed is a closed rectangular prism 4 m by 3 m by 2.5 m (no floor — that's concrete). Calculate the painter's quote.
2. Find the mistake
Another Year 10 student has tried to find the surface area of an open-top rectangular box with dimensions 20 cm × 15 cm × 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 the surface area of an open-top 20 × 15 × 10 cm box:
Line 1: Closed SA = 2(20×15 + 20×10 + 15×10) = 2(300 + 200 + 150) = 1300 cm²
Line 2: Open-top means we subtract the SIDE face: 20 × 10 = 200 cm²
Line 3: Open SA = 1300 − 200 = 1100 cm²
(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? "Open-top" means the top face is missing, not a side. The top is the face with dimensions length × width (20 × 15).3. Open-ended challenge — design the cheapest box
This question has many valid answers. Be creative but show every number. 4 marks
3.1 A small business needs a closed rectangular box with an internal volume of exactly 1000 cm³ (one litre). Cardboard costs $4 per m² (= $0.0004 per cm²).
Design two different boxes that both hold 1000 cm³:
(i) Box A: a long thin box (e.g., side lengths in the ratio 1 : 1 : 10 or similar).
(ii) Box B: as close to a cube as possible.
For each box you design:
(a) State the three integer dimensions in cm (the product must be 1000).
(b) Calculate the surface area in cm².
(c) Calculate the cardboard cost.
(d) Which design is cheaper? Write one sentence explaining why.
Hint: Integer triples that multiply to 1000 include (1, 1, 1000), (1, 10, 100), (4, 5, 50), (5, 10, 20), (10, 10, 10), (4, 10, 25), (2, 20, 25).
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Closed 14 × 9 × 6 box
SA = 2(14×9 + 14×6 + 9×6) = 2(126 + 84 + 54) = 2 × 264 = 528 cm².
1.2 — Cube side 9 cm
SA = 6 × 9² = 6 × 81 = 486 cm².
1.3 — Open-top crate 60 × 40 × 30 cm
Closed SA = 2(60×40 + 60×30 + 40×30) = 2(2400 + 1800 + 1200) = 10 800 cm². Subtract top = 60 × 40 = 2400 cm². Wood = 8400 cm².
1.4 — Triangular prism (base 10, ht 6, third side ≈ 11.66, length 18)
Two triangles = 2 × ½ × 10 × 6 = 60 cm².
Three rectangles = 10 × 18 + 6 × 18 + 11.66 × 18 = 180 + 108 + 209.88 = 497.88 cm².
Total SA = 60 + 497.88 = 557.88 cm² (≈ 558 cm²).
The third side 11.66 cm is used for the third rectangle, not for the triangle area.
1.5 — Closed display case 1.5 × 0.6 × 0.4 m
SA = 2(1.5×0.6 + 1.5×0.4 + 0.6×0.4) = 2(0.9 + 0.6 + 0.24) = 2 × 1.74 = 3.48 m².
Cost = 3.48 × $95 = $330.60.
1.6 — Shed 4 × 3 × 2.5 m, no floor
Closed SA = 2(4×3 + 4×2.5 + 3×2.5) = 2(12 + 10 + 7.5) = 2 × 29.5 = 59 m². Subtract floor = 4 × 3 = 12 m². Painted area = 47 m².
Quote = 47 × $7 = $329.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) "Open-top" means the top face is missing, not a side. The top face has dimensions length × width = 20 × 15, not 20 × 10 (which is a side face).
(c) Corrected working:
Closed SA = 2(20×15 + 20×10 + 15×10) = 1300 cm² ✓
Subtract missing top = 20 × 15 = 300 cm²
Open SA = 1300 − 300 = 1000 cm².
Always identify which face is missing before subtracting. "Top" = the lw face.
3 — Open-ended challenge (sample solutions)
We need two closed boxes, each with volume 1000 cm³, and compare the surface areas.
Box A — long thin (5, 10, 20)
Check volume: 5 × 10 × 20 = 1000 cm³ ✓
SA = 2(5×10 + 5×20 + 10×20) = 2(50 + 100 + 200) = 700 cm².
Cost = 700 × $0.0004 = $0.28.
Box B — close to a cube (10, 10, 10)
Check volume: 10 × 10 × 10 = 1000 cm³ ✓
SA = 6 × 10² = 600 cm².
Cost = 600 × $0.0004 = $0.24.
(d) Box B (the cube) is cheaper. A cube has the minimum surface area for any given volume of a rectangular prism — the more "stretched-out" the box, the more cardboard it needs to enclose the same volume.
Marking: 1 for two valid (1000 cm³) triples; 1 for correctly computed surface area on each box; 1 for correct cost on each box; 1 for the comparison sentence identifying the cube as cheaper and noting the cube-min-SA principle. Other valid pairs accepted.