Mathematics • Year 9 • Unit 4 • Lesson 18
Surface Area, Volume and Scale Factors
Build fluency with V = πr²h, prism volumes, surface area, and the scale rules: length scales by k, area by k², volume by k³. One worked example, one faded example, then eight graduated practice problems.
1. I do — fully worked example
Two parts: the cylinder formulas, then a scale-factor application.
Problem. A cylinder has radius 4 cm and height 10 cm. (a) Find the volume. (b) Find the surface area. (c) A similar cylinder is built with all lengths tripled. Find its volume using a scale factor.
Step 1 — Recall the formulas.
V = π × r² × h
SA = 2πr² + 2πrh (two circular ends + curved side)
Reason: lesson § "Area and Volume Formulas".
Step 2 — Substitute r = 4, h = 10 into V.
V = π × 4² × 10 = π × 16 × 10 = 160π cm³ ≈ 502.7 cm³
Step 3 — Substitute into SA.
SA = 2π(4²) + 2π(4)(10) = 32π + 80π = 112π cm² ≈ 351.9 cm²
Step 4 — Apply the scale factor k = 3 to volume.
Volume scales by k³ = 3³ = 27.
New volume = 27 × 160π = 4 320π cm³ ≈ 13 571.7 cm³.
Reason: lesson § "Scale Factors" — volume scales as k³, not k.
Answer: V ≈ 502.7 cm³; SA ≈ 351.9 cm²; new V = 4 320π ≈ 13 571.7 cm³.
2. We do — fill in the missing steps
A similar-cone problem from the lesson. Fill the blanks. 4 marks
Problem. Two similar cones have heights 5 cm and 15 cm. The smaller has volume 20π cm³. Find the volume of the larger cone.
Step 1 — Find the length scale factor k.
k = ____ ÷ ____ = ______
Step 2 — Volume scales by k to the power of ____.
Step 3 — Multiplier for volume.
k³ = ____³ = ______
Step 4 — Large volume.
V_large = ______ × 20π = ________ cm³
3. You do — independent practice
Foundation = single formula. Standard = combine two formulas. Extension = composite or reverse-engineer.
Foundation — single formula
3.1 Find the volume of a rectangular prism 4 cm × 3 cm × 5 cm. 1 mark
3.2 Find the surface area of a rectangular prism 4 cm × 3 cm × 2 cm. 1 mark
3.3 Find the volume of a cylinder with radius 3 cm and height 4 cm. Leave answer in terms of π. 1 mark
3.4 A scale factor for length is k = 2. By what factor does area scale? By what factor does volume scale? 1 mark
Standard — combine two ideas
3.5 Two similar solids have lengths in ratio 2 : 5. Find the ratio of their (a) surface areas, (b) volumes. 2 marks
3.6 A model car is 1 : 24 scale. The real car has volume 4.5 m³. Find the volume of the model in cm³. (Hint: 1 m³ = 1 000 000 cm³.) 2 marks
Extension — push your thinking
3.7 Find the volume of a composite solid: a rectangular prism 6 cm × 5 cm × 4 cm with a cylindrical hole drilled through (radius 1 cm, depth = the 4 cm height). Leave answer to 1 d.p. 3 marks
3.8 Two similar pyramids have base areas 16 cm² and 64 cm². (a) Find the length scale factor k. (b) Find the ratio of their volumes. (c) If the smaller pyramid has volume 50 cm³, find the larger volume. 3 marks
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What I'll revisit before next class:
Section 2 — We do (similar cones)
Step 1: k = 15 ÷ 5 = 3.
Step 2: volume scales by k to the power of 3.
Step 3: k³ = 3³ = 27.
Step 4: V_large = 27 × 20π = 540π cm³ ≈ 1 696.5 cm³.
3.1 — Prism 4×3×5
V = 4 × 3 × 5 = 60 cm³.
3.2 — Prism 4×3×2 SA
SA = 2(lw + lh + wh) = 2(12 + 8 + 6) = 2 × 26 = 52 cm².
3.3 — Cylinder r=3, h=4
V = π × 3² × 4 = 36π cm³ ≈ 113.1 cm³.
3.4 — k = 2 scaling
Area scales by k² = 4×. Volume scales by k³ = 8×.
3.5 — Lengths 2 : 5
(a) Area ratio = 2² : 5² = 4 : 25.
(b) Volume ratio = 2³ : 5³ = 8 : 125.
3.6 — Model car 1:24
Volume scale factor = 24³ = 13 824.
Model volume = 4.5 ÷ 13 824 m³ ≈ 3.255 × 10⁻⁴ m³.
In cm³: 3.255 × 10⁻⁴ × 1 000 000 ≈ 325.5 cm³.
3.7 — Prism with cylindrical hole
Prism volume = 6 × 5 × 4 = 120 cm³.
Cylinder removed = π × 1² × 4 = 4π ≈ 12.57 cm³.
Remaining volume = 120 − 12.57 ≈ 107.4 cm³.
3.8 — Similar pyramids
(a) Area ratio = 64/16 = 4, so k² = 4, k = 2.
(b) Volume ratio = k³ = 2³ = 8 : 1 (large : small).
(c) V_large = 8 × 50 = 400 cm³.