← Unit 4 Similarity and Scale Factors

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Similarity and Scale Factors

⏱ 25 min📚 Year 9📈
Think First

Two similar rectangles have sides in ratio 2:3. If the smaller has area 24 cm², what is the area of the larger?

💡 Revision: Ensure you understand ratios and can work with proportions.

Learning Intentions

Know

  • Similar figures
  • Scale factor
  • Area scale factor
  • Volume scale factor

Understand

  • Why area scales as the square of the linear scale factor
  • Why volume scales as the cube

Can Do

  • Identify similar figures
  • Calculate scale factors
  • Find areas and volumes of similar solids
Scale factorSimilarEnlargementReduction
Learn Phase
1

Similar Figures

Same shape, different size

Two figures are similar if:

  • Corresponding angles are equal
  • Corresponding sides are in proportion

The scale factor ($k$) is the ratio of corresponding sides:

Scale Factor

$k = dfrac{\text{new length}}{\text{original length}}$

If $k > 1$: enlargement. If $0 < k < 1$: reduction.

2

Area and Scale Factor

Area scales as $k^2$

If lengths scale by factor $k$, then:

Area Scale Factor

$\text{New Area} = k^2 \times \text{Original Area}$

Example: Scale factor 3. A rectangle 2×4 (area 8) becomes 6×12 (area 72).

$72 = 3^2 \times 8 = 9 \times 8$ ✓

This applies to all 2D shapes, not just rectangles.

3

Volume and Scale Factor

Volume scales as $k^3$

For 3D solids, if lengths scale by $k$:

Volume Scale Factor

$\text{New Volume} = k^3 \times \text{Original Volume}$

Example: A cube of side 2 (volume 8) is enlarged by scale factor 3.

New side = 6, new volume = 216.

$216 = 3^3 \times 8 = 27 \times 8$ ✓

✓

Check Understanding

Try it yourself

Two similar cylinders have radii in ratio 1:2. If the smaller has volume 10 cm³, find the larger volume.

Worked Example

Similarity and Scale

1

Two similar triangles have sides in ratio 2:5. The smaller has area 12 cm². Find the larger area.

$k = 5/2 = 2.5$

$\text{Area} = k^2 \times 12 = 6.25 \times 12 = 75$ cm²

2

A model car is built at scale 1:24. If the real car is 4.8 m long, how long is the model?

$\text{Model length} = 4.8 / 24 = 0.2$ m = 20 cm

3

A sphere of radius 3 cm has volume $36pi$ cm³. A similar sphere has radius 6 cm. Find its volume.

$k = 6/3 = 2$

$V = 2^3 \times 36pi = 8 \times 36pi = 288pi$ cm³

Common Misconceptions

✕

Area scales linearly with scale factor. No, area scales as $k^2$. If lengths double, area quadruples.

✕

Volume scales as $k^2$. No, volume scales as $k^3$. If lengths double, volume increases 8-fold.

✕

Scale factor applies directly to angles. No, angles in similar figures are equal, not scaled.

Your Turn

Practice, Scale Factors

Work through each question in your book or digitally. Answers are in the Questions phase.

1Two similar rectangles have sides 3:4. The smaller has area 27 cm². Find the larger area.
2A map has scale 1:50,000. A park is 2 cm × 3 cm on the map. What is its actual area in km²?
3A model building is 1:100 scale. If the model volume is 0.5 m³, what is the actual volume?
Real-World Anchor

Cartography and Modelling

Maps use scale factors extensively. A 1:25,000 map means 1 cm on the map equals 250 m in reality. Architects build scale models where area and volume calculations inform material estimates for the full-sized structure.

📓 Copy Into Your Books

▼

Similar

  • Equal angles
  • Proportional sides
  • Scale factor $k$

Area

  • Scales as $k^2$
  • $A_{\text{new}} = k^2 A_{\text{old}}$

Volume

  • Scales as $k^3$
  • $V_{\text{new}} = k^3 V_{\text{old}}$
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Scale factor 3. Area becomes:
Scale factor 2. Volume becomes:
Two similar triangles, sides 2:3. Smaller area = 8. Larger area:
Scale 1:50,000. 2 cm on map = real distance:
Similar figures have:
Cube side tripled. Surface area:
Model 1:10. Real volume is model volume ×
Map scale 1:100,000. 1 cm² on map = real area:
1Two similar cones have heights 4 cm and 10 cm. The smaller has volume 20 cm³. Find the larger volume.
2A photograph is enlarged by scale factor 1.5. If the original area is 80 cm², find the new area.
3Explain why volume scales as $k^3$ when lengths scale as $k$.

Comprehensive Answers

1Similar cones heights 4 and 10. Smaller V=20. Larger V?
$k = 10/4 = 2.5$. $V = 2.5^3 \times 20 = 15.625 \times 20 = 312.5$ cm³.
2Photo enlarged 1.5×. Original area 80 cm².
$A = 1.5^2 \times 80 = 2.25 \times 80 = 180$ cm².
3Why volume scales as $k^3$.
Volume = length × width × height. Each dimension scales by $k$, so $k \times k \times k = k^3$.
MC 1Scale 3, area.
$3^2 = 9$. Answer: C
MC 2Scale 2, volume.
$2^3 = 8$. Answer: D
MC 3Triangles 2:3, area 8.
$k=1.5$, $A = 2.25 \times 8 = 18$. Answer: C
MC 41:50,000, 2 cm.
$2 \times 50,000 = 100,000$ cm = 1 km. Answer: B
MC 5Similar figures.
Equal angles, proportional sides. Answer: B
MC 6Cube side tripled, SA.
$3^2 = 9$. Answer: C
MC 7Model 1:10, volume.
$10^3 = 1000$. Answer: C
MC 81:100,000, 1 cm².
$(1 \times 100,000)^2$ cm² = $10^{10}$ cm² = 1 km². Answer: C
SA 1Cones heights 4 and 10.
$312.5$ cm³.
SA 2Photo enlarged.
$180$ cm².
SA 3Volume scales $k^3$.
Three dimensions each scale by $k$.
Game Phase
🎲
Game Unlocked!
You have mastered the Check Your Understanding questions. Choose a game mode below.
📦
Classify & Sort
Sort mathematical objects by their properties.
⚡
Speed Challenge
Answer questions against the clock.
📈
Match Maker
Match problems to their solutions.
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