Mathematics • Year 10 • Unit 3 • Lesson 13
Area and Volume Ratios — Mixed Challenge
Stitch every Lesson 13 idea together: forwards (given k, find new area or volume), backwards (given area or volume ratio, find k), spot the classic "scale-factor-as-multiplier vs adder" mistake, and design your own pair of similar shapes with a specified volume ratio.
1. Mixed problems
Each problem pulls on a different idea from Lesson 13. Decide whether to use k, k² or k³ before you start. 2-3 marks each
1.1 Two similar cones have radii 3 cm and 12 cm. The smaller cone has a curved surface area of 30 cm². Find the curved surface area of the larger cone. 3 marks
1.2 A model railway is built at a scale of 1:87 (HO scale). A locomotive in real life is 17.4 m long and contains 8,700 L of water in its tender. Find the model's length in cm and the model's water capacity in mL. 3 marks
1.3 Two similar swimming pools have volumes 8,000 L and 27,000 L. The smaller pool needs 100 m² of paint to coat its inside surface. Find the area of paint needed for the larger pool. 3 marks
1.4 A scale plan of a basketball court (28 m × 15 m in real life) is to be drawn so that the court fits exactly inside a rectangle of 14 cm × 7.5 cm on a sheet of paper. State the scale (1 : ?) and the area scale factor. 3 marks
1.5 Two similar prisms have surface areas in the ratio 4 : 25. Find the ratio of their volumes (small : large). 2 marks
1.6 A small ice-cream cone has a height of 6 cm and holds 30 mL of ice cream. A similar large cone is to be made that holds 240 mL. Find the height of the large cone. 3 marks
2. Find the mistake
Another Year 10 student has tried to find the volume of a larger similar prism. Their working is shown. Exactly one line contains a mistake. Spot it, explain why it is wrong, and re-do the working correctly. 3 marks
Problem the student tried: Two similar prisms have linear scale factor k = 3. The smaller prism has a volume of 50 cm³. Find the volume of the larger prism.
Line 1: k = 3.
Line 2: Volume scale factor = k² = 9.
Line 3: V_large = 50 × 9 = 450 cm³.
Line 4: Final answer: 450 cm³.
(a) Which line contains the mistake?
(b) Explain in one or two sentences why that line is wrong.
(c) Write out the corrected line and give the correct final answer.
Stuck? Volume depends on three dimensions, not two.3. Open-ended challenge — design with a target volume ratio
This question has many valid answers. Be creative but show every number. 4 marks
3.1 Design two different rectangular prisms (boxes) with these properties:
- both have integer side lengths,
- they are similar to each other,
- the larger box has volume exactly 64 times the smaller box.
For each prism, state the dimensions (length × width × height), the surface area, and the volume. Then state the linear scale factor k, the area scale factor k², and verify k³ = 64.
Bonus: State what the ratio of the two surface areas is, and confirm it equals k².
How did this worksheet feel?
What I'll revisit before next class:
1.1 — Cones, radii 3 and 12
k = 12/3 = 4. k² = 16.
Larger curved surface area = 30 × 16 = 480 cm².
1.2 — HO scale 1:87 locomotive
k = 1/87.
Model length = 17.4 m ÷ 87 = 0.2 m = 20 cm.
Volume scale factor = (1/87)³ = 1/658,503.
Model water capacity = 8700 L ÷ 658,503 ≈ 0.01322 L ≈ 13.2 mL.
1.3 — Swimming pools, volumes 8,000 and 27,000
Volume ratio = 27000 / 8000 = 27/8.
k = ∛(27/8) = 3/2. k² = 9/4.
Paint for larger pool = 100 × (9/4) = 225 m².
1.4 — Basketball court plan
Real length = 28 m = 2800 cm. Plan length = 14 cm. Scale = plan : real = 14 : 2800 = 1 : 200.
Check using the width: 7.5 cm : 1500 cm = 1 : 200 ✓.
Area scale factor (real / plan) = 200² = 40,000.
1.5 — Surface areas 4:25
k² = 25/4, so k = 5/2.
Volume ratio = k³ = 125/8, i.e. small : large = 8 : 125.
1.6 — Ice-cream cones
Volume ratio = 240 / 30 = 8.
k = ∛8 = 2.
Large cone height = 6 × 2 = 12 cm.
2 — Find the mistake
(a) The mistake is on Line 2.
(b) For similar prisms, the volume scale factor is k³, not k². Volume depends on three dimensions (length × width × height), so all three get multiplied by k → k³ in total. Squaring would give the area scale factor, not the volume scale factor.
(c) Corrected: Volume scale factor = k³ = 3³ = 27. V_large = 50 × 27 = 1350 cm³.
3 — Open-ended (sample solution)
k = ∛64 = 4, so every side of the larger box must be 4× the corresponding side of the smaller one.
Box A (smaller): 1 cm × 2 cm × 3 cm. Volume = 6 cm³. Surface area = 2(1×2 + 1×3 + 2×3) = 2(2+3+6) = 22 cm².
Box B (larger): 4 cm × 8 cm × 12 cm (every side × 4). Volume = 4 × 8 × 12 = 384 cm³. Surface area = 2(4×8 + 4×12 + 8×12) = 2(32+48+96) = 352 cm².
Volume ratio = 384 / 6 = 64 = k³ ✓.
Bonus: Surface area ratio = 352 / 22 = 16 = k² ✓.
Marking: 1 for correctly deriving k = 4, 1 for a valid small box (any integer dimensions), 1 for the corresponding large box with all sides ×4, 1 for verifying both k³ = 64 and k² = 16 with the numbers.