← 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

$k = dfrac{ ext{new length}}{ ext{original length}}$

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

$ ext{New Area} = k^2 imes ext{Original Area}$

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

$72 = 3^2 imes 8 = 9 imes 8$ ✓

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

$ ext{New Volume} = k^3 imes ext{Original Volume}$

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

New side = 6, new volume = 216.

$216 = 3^3 imes 8 = 27 imes 8$ ✓

✓

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.

$k = 5/2 = 2.5$

$ ext{Area} = k^2 imes 12 = 6.25 imes 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?

$ ext{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.

$k = 6/3 = 2$

$V = 2^3 imes 36pi = 8 imes 36pi = 288pi$ 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 $k^2$
  • $A_{ ext{new}} = k^2 A_{ ext{old}}$

Volume

  • Scales as $k^3$
  • $V_{ ext{new}} = k^3 V_{ ext{old}}$
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?
$k = 10/4 = 2.5$. $V = 2.5^3 imes 20 = 15.625 imes 20 = 312.5$ cm³.
2Photo enlarged 1.5×. Original area 80 cm².
$A = 1.5^2 imes 80 = 2.25 imes 80 = 180$ cm².
3Why volume scales as $k^3$.
Volume = length × width × height. Each dimension scales by $k$, so $k imes k imes k = k^3$.
MC 1Scale 3, area.
$3^2 = 9$. Answer: C
MC 2Scale 2, volume.
$2^3 = 8$. Answer: D
MC 3Triangles 2:3, area 8.
$k=1.5$, $A = 2.25 imes 8 = 18$. Answer: C
MC 41:50,000, 2 cm.
$2 imes 50,000 = 100,000$ 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.
$(1 imes 100,000)^2$ 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.