Mathematics • Year 10 • Unit 3 • Lesson 13
Area and Volume Ratios — Skill Drill
Build fluency moving between the three scale factors from Lesson 13: linear k, area k², volume k³. Work forwards (given k, find the area or volume change), and backwards (given an area or volume ratio, find k = √ratio or k = ∛ratio).
1. I do — fully worked example
Read every step. Each one has a short reason on the right so you can see why, not just what.
Problem. Two similar cylinders have radii in the ratio 2 : 5. The smaller cylinder has a volume of 32 cm³. Find the volume of the larger cylinder.
Step 1 — Identify the linear scale factor k.
k = 5 / 2 (large ÷ small)
Reason: scale factor is the ratio of corresponding lengths, large over small.
Step 2 — Find the volume scale factor k³.
k³ = (5/2)³ = 125 / 8
Reason: volume scales by the cube of the linear scale factor.
Step 3 — Multiply the small volume by k³.
V_large = 32 × 125/8 = 4 × 125 = 500 cm³
Reason: V_large / V_small = k³, so V_large = V_small × k³.
Answer: V_large = 500 cm³.
2. We do — fill in the missing steps
Same structure as Section 1, but with the working faded. Fill in each blank. 5 marks
Problem. A map is drawn to a scale of 1:50,000. A rectangular paddock measures 4 cm by 3 cm on the map. Find the actual area of the paddock in km².
Step 1 — State the linear scale factor (real / map):
k = ______
Step 2 — Convert each map dimension to real-world length:
Real length = 4 cm × ______ = ______ cm = ______ m = ______ km
Real width = 3 cm × ______ = ______ cm = ______ m = ______ km
Step 3 — Calculate the real area:
Real area = ______ × ______ = ______ km²
Step 4 — Verify with k²:
Map area = 4 × 3 = 12 cm². k² = ______² . Real area = 12 × ______ = ______ cm² = ______ km². ✓
3. You do — independent practice
Show your working. The first four are foundation, the middle two standard, the last two extension.
Foundation — k, k², k³ both ways
3.1 Linear scale factor k = 3. Find the area scale factor and the volume scale factor. 1 mark
3.2 Linear scale factor k = 1/2. Find the area scale factor and the volume scale factor. 1 mark
3.3 The area scale factor between two similar figures is 36. Find k. 1 mark
3.4 The volume scale factor between two similar figures is 64. Find k. 1 mark
Standard — combine k, k² and k³
3.5 Two similar spheres have radii 2 cm and 6 cm. The smaller sphere has volume 32 cm³. Find the volume of the larger sphere. 2 marks
3.6 Two similar paintings have linear scale factor k = 5/2. The smaller painting has area 40 cm². Find the area of the larger painting. 2 marks
Extension — work backwards
3.7 Two similar prisms have volumes 27 cm³ and 1000 cm³. Find the linear scale factor (large / small) and the area scale factor. 3 marks
3.8 A model car is built to a scale of 1:24. Its fuel tank in the real car holds 48 L. Find the model fuel tank's capacity in mL. 2 marks
How did this worksheet feel?
What I'll revisit before next class:
Section 2 — We do (Map 1:50,000)
Step 1: k = 50,000.
Step 2: Real length = 4 × 50,000 = 200,000 cm = 2,000 m = 2 km. Real width = 3 × 50,000 = 150,000 cm = 1,500 m = 1.5 km.
Step 3: Real area = 2 × 1.5 = 3 km².
Step 4: Map area = 12 cm². k² = 50,000² = 2.5 × 10⁹. Real area in cm² = 12 × 2.5 × 10⁹ = 3 × 10¹⁰ cm² = 3 km² ✓.
3.1 — k = 3
k² = 9. k³ = 27.
3.2 — k = 1/2
k² = 1/4. k³ = 1/8.
3.3 — Area scale factor 36
k² = 36, so k = √36 = 6.
3.4 — Volume scale factor 64
k³ = 64, so k = ∛64 = 4.
3.5 — Spheres 2 cm and 6 cm
k = 6/2 = 3. k³ = 27.
V_large = 32 × 27 = 864 cm³.
3.6 — Paintings, k = 5/2
k² = 25/4.
A_large = 40 × (25/4) = 10 × 25 = 250 cm².
3.7 — Prisms with volumes 27 and 1000
Volume ratio = 1000 / 27. k = ∛(1000/27) = 10/3.
Area scale factor = k² = (10/3)² = 100/9.
3.8 — Model car, 1:24
k = 1/24. Volume scale factor = (1/24)³ = 1/13,824.
Model capacity = 48 ÷ 13,824 ≈ 0.00347 L ≈ 3.47 mL.