Mathematics • Year 10 • Unit 1 • Lesson 16

Surface Area of Prisms in the Real World

Apply rectangular and triangular prism surface area to real Australian situations — packaging quotes, paint coverage, wood for planter boxes and glass for fish tanks. Then explain your method in your own words.

Apply · Real-World Maths

1. Word problems

Each problem uses the prism surface-area methods from Lesson 16: SA = 2(lw + lh + wh) for a closed rectangular prism, "subtract any missing face" for an open prism, or "2 triangles + 3 rectangles" for a triangular prism. Show your working — a final answer with no working only earns half marks.

1.1 — Painting a closed storage container. A closed shipping container measures 2.5 m by 1.8 m by 1.2 m. A painter quotes based on the total external surface area.

(a) Calculate the total surface area of the container.
(b) If 1 litre of paint covers 12 m², how many litres are needed for one coat? 3 marks

Stuck? Use SA = 2(lw + lh + wh), then divide by 12 to get the litres needed.

1.2 — Wooden planter box. A community garden builds an open-top planter box 1.2 m by 0.8 m by 0.6 m. Marine ply costs $48 per m².

(a) Calculate the area of ply needed (open top — no lid).
(b) Calculate the cost of the ply for one planter box. 3 marks

Stuck? Closed SA minus the missing top (1.2 × 0.8). Then multiply by $48.

1.3 — Triangular ridge tent. A camping tent is a triangular prism. The two ends are right-angled triangles with legs 1.8 m and 2.4 m (hypotenuse 3.0 m). The tent is 3.5 m long. The floor of the tent (the 1.8 m × 3.5 m rectangle) is canvas; the other two sloping walls (2.4 m × 3.5 m and 3.0 m × 3.5 m) and the two triangular ends are nylon.

(a) Calculate the area of canvas (floor only).
(b) Calculate the area of nylon (two triangles plus two sloping rectangles). 4 marks

Stuck? Floor area = 1.8 × 3.5. Each triangular end uses the perpendicular legs 1.8 and 2.4, not the hypotenuse 3.0.

1.4 — Cardboard packaging. A company manufactures open-top cardboard boxes for shipping. Each box measures 40 cm by 30 cm by 25 cm. Cardboard costs $3.50 per m².

(a) Calculate the area of cardboard needed for one box, in m².
(b) The company makes 500 boxes. Calculate the total cardboard cost. 3 marks

Stuck? Convert cm² to m² by dividing by 10 000 before multiplying by $3.50.

1.5 — Fish-tank glass quote. A custom fish tank is open at the top and measures 90 cm by 45 cm by 50 cm. The aquarium shop charges $0.85 per 100 cm² of 6 mm glass.

(a) Calculate the total area of glass needed.
(b) Calculate the cost of the glass. 3 marks

Stuck? Glass area = closed SA − top face (90 × 45). Cost = (area ÷ 100) × $0.85.

2. Explain your thinking

This question is about communication, not just numbers. Use full sentences. 4 marks

2.1 A friend says: "When I make a box bigger, the surface area and the volume both go up — so they're basically the same thing." Using a worked example from Lesson 16 (you may use the 30 cm × 20 cm × 10 cm shoebox), explain (i) how surface area and volume are actually different quantities, (ii) why they don't scale the same way when you double a dimension, and (iii) why a packaging engineer cares about surface area specifically when choosing how much cardboard to buy.

Stuck? Surface area is measured in cm² (covers the outside); volume is in cm³ (fills the inside). Try doubling 30 × 20 × 10 to 60 × 40 × 20 and see how each changes.

How did this worksheet feel?

Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

1.1 — Closed shipping container 2.5 × 1.8 × 1.2 m

(a) SA = 2(2.5×1.8 + 2.5×1.2 + 1.8×1.2) = 2(4.5 + 3 + 2.16) = 2 × 9.66 = 19.32 m².
(b) Litres = 19.32 ÷ 12 ≈ 1.61 ⇒ 2 litres needed (round up — paint sold in whole litres).

1.2 — Open-top planter box 1.2 × 0.8 × 0.6 m

(a) Closed SA = 2(1.2×0.8 + 1.2×0.6 + 0.8×0.6) = 2(0.96 + 0.72 + 0.48) = 4.32 m². Subtract top = 1.2 × 0.8 = 0.96 m². Open SA = 3.36 m².
(b) Cost = 3.36 × $48 = $161.28.

1.3 — Triangular ridge tent

(a) Canvas (floor only) = 1.8 × 3.5 = 6.3 m².
(b) Two triangles = 2 × ½ × 1.8 × 2.4 = 4.32 m². Two sloping rectangles = 2.4 × 3.5 + 3.0 × 3.5 = 8.4 + 10.5 = 18.9 m². Total nylon = 4.32 + 18.9 = 23.22 m².
Triangle area uses the perpendicular legs 1.8 and 2.4, not the hypotenuse 3.0.

1.4 — Cardboard packaging 40 × 30 × 25 cm, open top

(a) Closed SA = 2(40×30 + 40×25 + 30×25) = 2(1200 + 1000 + 750) = 5900 cm². Subtract top = 1200 cm². Open SA = 4700 cm² = 0.47 m².
(b) Cost per box = 0.47 × $3.50 = $1.645. Total = 500 × $1.645 = $822.50.

1.5 — Custom fish tank 90 × 45 × 50 cm, open top

(a) Closed SA = 2(90×45 + 90×50 + 45×50) = 2(4050 + 4500 + 2250) = 21 600 cm². Subtract top = 90 × 45 = 4050. Glass area = 17 550 cm².
(b) Cost = (17 550 ÷ 100) × $0.85 = 175.5 × $0.85 = $149.18 (to nearest cent).

2.1 — Explain your thinking (sample response)

(i) Surface area measures the outside skin of a 3D shape (in cm² or m²), while volume measures the inside space (in cm³ or m³). They are different quantities with different units.
(ii) When I double every dimension of the shoebox (30 × 20 × 10) to (60 × 40 × 20), the surface area goes from 2200 cm² to 2(2400 + 1200 + 800) = 8800 cm² (×4), but the volume goes from 6000 cm³ to 48 000 cm³ (×8). They do not scale the same way — surface area scales by the square of the linear factor, volume by the cube.
(iii) A packaging engineer cares about surface area because that is the amount of cardboard they have to buy. Volume tells them what fits inside, but it does not tell them how much material to order. Two boxes with the same volume can use very different amounts of cardboard.

Marking: 1 for naming the unit difference (cm² vs cm³), 1 for a correct numerical comparison after doubling, 1 for the "×4 vs ×8" relationship, 1 for the packaging engineer's reason in a full sentence.