Mathematics • Year 8 • Unit 3 • Lesson 9
Surface Area in the Real World
Use SA where it actually shows up: wrapping presents, painting walls, labelling cans, and lining swimming pools. Then explain when SA matters more than volume.
1. Word problems
Each problem hides a prism or cylinder. Sketch the shape, identify the dimensions, choose the right SA formula. A correct final answer without working earns only half marks.
1.1 — Wrapping a present. A rectangular gift box is 25 cm long, 15 cm wide, and 10 cm tall. Sara wants to wrap it in paper with no overlap (an ideal wrap).
(a) Find the SA of the box.
(b) A sheet of wrapping paper measures 70 cm × 50 cm. Will it be enough? Justify your answer numerically. 3 marks
1.2 — Painting walls and ceiling. A bedroom is a rectangular prism 4 m long, 3 m wide, and 2.5 m tall. The owner will paint the four walls and the ceiling (but NOT the floor).
(a) Find the area of the four walls (lateral area).
(b) Find the area of the ceiling.
(c) Total area to paint? 3 marks
1.3 — Tin can label. A baked-beans tin is a cylinder with radius 3.5 cm and height 11 cm. The label wraps around the curved surface only (top and bottom are not covered).
(a) Find the area of paper needed for the label (i.e. the curved surface area).
(b) The label is sold in rolls 12 cm wide. What length of label is unrolled around the can? (Hint: the unrolled label is a rectangle of width = 2πr and height = h.) 3 marks
1.4 — Pool tiling. A rectangular swimming pool is 12 m long, 6 m wide, and 2 m deep. The owner tiles the FOUR walls and the FLOOR (the surface of the water is open, so no top).
(a) Find the area of the floor.
(b) Find the area of the four walls.
(c) Total tile area? Tiles cost $45/m². What is the total tiling cost? 3 marks
1.5 — Soup-can manufacturer. A factory makes sealed cylindrical soup cans with radius 4 cm and height 12 cm. Each can needs tin sheet for the full SA (2 circles + curved surface).
(a) Find the SA of one can in exact form (nπ) and as a decimal (nearest cm²).
(b) The factory makes 10 000 cans per day. How many square metres of tin sheet are needed per day? (Hint: 10 000 cm² = 1 m².) 3 marks
2. Explain your thinking
This question is about communication, not just answers. Use full sentences. 4 marks
2.1 A classmate is asked to find the SA of a cylindrical can with diameter 10 cm and height 14 cm. They write "SA = 2π(10)² + 2π(10)(14) = 200π + 280π = 480π ≈ 1508 cm²". In your own words, explain (i) what mistake they have made, (ii) what the correct answer is and how to get it, and (iii) one quick "sanity check" that would have warned them their answer was wrong. Use the phrase "diameter is NOT radius" somewhere in your answer.
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Wrapping a present
(a) SA = 2(25×15 + 25×10 + 15×10) = 2(375 + 250 + 150) = 2 × 775 = 1550 cm².
(b) Paper area = 70 × 50 = 3500 cm². Since 3500 > 1550, yes — the paper is enough (with 1950 cm² to spare).
1.2 — Painting bedroom
(a) Four walls = 2(4 × 2.5) + 2(3 × 2.5) = 20 + 15 = 35 m².
(b) Ceiling = 4 × 3 = 12 m².
(c) Total = 35 + 12 = 47 m² to paint.
1.3 — Tin can label
(a) CSA = 2π × 3.5 × 11 = 77π ≈ 241.9 cm².
(b) Length unrolled = circumference = 2π × 3.5 = 7π ≈ 22.0 cm. (Width of label = height of can = 11 cm.)
1.4 — Pool tiling
(a) Floor = 12 × 6 = 72 m².
(b) Walls = 2(12 × 2) + 2(6 × 2) = 48 + 24 = 72 m².
(c) Total = 72 + 72 = 144 m². Cost = 144 × $45 = $6 480.
1.5 — Soup-can manufacturer
(a) SA = 2π(16) + 2π(4)(12) = 32π + 96π = 128π ≈ 402 cm².
(b) 10 000 × 402 = 4 020 000 cm² = 402 m² of tin sheet per day.
2.1 — Explain your thinking (sample response)
The classmate has used the diameter (10 cm) instead of the radius in the SA formula — but diameter is NOT radius. The radius is half the diameter, so r = 10 / 2 = 5 cm. The correct working is: SA = 2π(5)² + 2π(5)(14) = 50π + 140π = 190π ≈ 596.9 cm². A quick sanity check: a 10 cm diameter can is much smaller than a 10 cm radius can (whose diameter would be 20 cm — wider than most household cans!), so the answer 1508 cm² is more than double what it should be — a warning sign.
Marking: 1 mark for spotting "used d not r"; 1 mark for showing r = 5 and correct working; 1 mark for the correct final answer (190π ≈ 596.9 cm²); 1 mark for the sanity check and clear use of "diameter is NOT radius".