Mathematics • Year 9 • Unit 4 • Lesson 5

Volume and Surface Area in the Real World

Use volume and surface area in real Aussie contexts: a soup can, a backyard pool, a chocolate box, a water tank, and a piece of timber. Then explain your method in your own words.

Apply · Real-World Maths

1. Word problems

Each problem uses V or SA of a prism or cylinder. Read carefully — some give diameter (halve it!), some ask for litres (1000 cm³ = 1 L), some need only one formula. Show your working. Round final answers to 1 dp unless told otherwise.

1.1 — Soup can. A cylindrical soup can has a diameter of 7 cm and a height of 11 cm.

(a) How much soup (in cm³, to 1 dp) does it hold when full?
(b) The factory needs to know how much metal (the total surface area, in cm²) is used per can. Find it to 1 dp. 3 marks

Stuck? Diameter 7 cm → radius 3.5 cm. V = π r² h, SA = 2π r² + 2π r h.

1.2 — Backyard pool. A rectangular swimming pool is 8 m long, 4 m wide, with a uniform depth of 1.5 m.

(a) Find the pool's volume in cubic metres.
(b) Convert to LITRES. (1 m³ = 1000 L.)
(c) The pool walls and floor need to be tiled. Find the total surface area to be tiled (the FOUR walls plus the floor — the top is open, no ceiling). 3 marks

Stuck on (c)? A pool has FIVE faces to tile (not 6). Add: 2 long walls + 2 short walls + floor.

1.3 — Chocolate box. A box of chocolates is a rectangular prism 24 cm long, 15 cm wide and 4 cm deep.

(a) Find the volume of the box.
(b) Find the total surface area (all 6 faces).
(c) The chocolates inside take up exactly 3/4 of the box's volume. What volume of chocolates is that? 3 marks

Stuck on (c)? Multiply the volume from (a) by 0.75.

1.4 — Water tank. A cylindrical rainwater tank has a diameter of 2 m and a height of 1.8 m.

(a) Find the tank's volume in cubic metres to 2 dp.
(b) Convert to LITRES (1 m³ = 1000 L).
(c) If a household uses 200 L of water per day, how many days will a full tank last? 3 marks

Stuck? Diameter 2 m → radius 1 m. V = π × 1² × 1.8 = 1.8π m³.

1.5 — Piece of timber. A length of structural pine timber has a square cross-section 90 mm × 90 mm and is 3 m long.

(a) Convert all measurements to the same units (cm), then find the volume in cm³.
(b) Find the volume in cubic metres (1 m³ = 1,000,000 cm³).
(c) The timber needs to be coated with a wood-stain that covers 8 m² per litre. Find the total surface area of the timber (in m²) and the volume of stain (in litres) needed. 3 marks

Stuck? 90 mm = 9 cm; 3 m = 300 cm. So cross-section area = 9 × 9 cm² and length = 300 cm. For (c) convert SA back to m² before dividing by 8.

2. Explain your thinking

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

2.1 A classmate writes: "I found the volume of a cylinder with diameter 10 cm and height 8 cm: V = π × 10² × 8 = 800π ≈ 2513 cm³." In your own words, explain (i) the mistake they made about diameter vs radius, (ii) what the correct radius is and what the correct volume should be, and (iii) by what FACTOR their answer is too big. Refer to "always use radius" somewhere in your explanation.

Stuck? Their answer used r = 10. The correct r is half — what's that? Then compare with the correct V.

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Answers — Do not peek before attempting

1.1 — Soup can (d = 7 cm, h = 11 cm)

Radius r = 3.5 cm.
(a) V = π × 3.5² × 11 = π × 12.25 × 11 = 134.75π ≈ 423.3 cm³.
(b) SA = 2π(3.5)² + 2π(3.5)(11) = 24.5π + 77π = 101.5π ≈ 318.9 cm².

1.2 — Backyard pool (8 × 4 × 1.5 m)

(a) V = 8 × 4 × 1.5 = 48 m³.
(b) 48 m³ × 1000 = 48,000 L.
(c) Floor = 8 × 4 = 32 m². Two long walls = 2 × (8 × 1.5) = 24 m². Two short walls = 2 × (4 × 1.5) = 12 m². Total to tile = 32 + 24 + 12 = 68 m².
Note: the open top doesn't count (no ceiling on a pool).

1.3 — Chocolate box (24 × 15 × 4 cm)

(a) V = 24 × 15 × 4 = 1440 cm³.
(b) SA = 2(24×15 + 24×4 + 15×4) = 2(360 + 96 + 60) = 2(516) = 1032 cm².
(c) Chocolates volume = 1440 × 3/4 = 1080 cm³.

1.4 — Water tank (d = 2 m, h = 1.8 m)

Radius r = 1 m.
(a) V = π × 1² × 1.8 = 1.8π ≈ 5.65 m³.
(b) 5.65 m³ × 1000 ≈ 5654.9 L (or ~5655 L).
(c) 5654.9 / 200 ≈ 28 days (just under 4 weeks).

1.5 — Piece of timber (90 mm × 90 mm × 3 m)

Convert: 9 cm × 9 cm × 300 cm.
(a) V = 9 × 9 × 300 = 24,300 cm³.
(b) V = 24,300 / 1,000,000 = 0.0243 m³.
(c) SA in cm²: 2(9×9 + 9×300 + 9×300) = 2(81 + 2700 + 2700) = 2(5481) = 10,962 cm². Convert to m²: 10,962 / 10,000 ≈ 1.10 m². Stain needed = 1.10 / 8 ≈ 0.14 L (about 140 mL).

2.1 — Explain your thinking (sample response)

My classmate used the DIAMETER (10 cm) in the formula instead of the RADIUS, which gives an answer that is far too big. The cylinder volume formula V = π r² h always uses radius, not diameter — so when a problem gives the diameter (here, 10 cm), you must halve it first to get r = 5 cm. The correct calculation is V = π × 5² × 8 = π × 25 × 8 = 200π ≈ 628.3 cm³, not 2513 cm³. The student's answer is too big by a factor of 4, because squaring the diameter instead of the radius means (2r)² = 4r² — exactly 4 times too much. A quick check: their answer of 2513 ÷ 628.3 ≈ 4. ✓

Marking: 1 mark for identifying diameter vs radius mistake; 1 for the correct radius (5); 1 for the correct V (200π or ≈628.3 cm³); 1 for explaining the factor of 4 too big.