Same shape, different size. Learn how scale factors connect lengths, areas, and volumes -- and why a scale factor of 2 makes the area four times bigger.
Today's hook: A model car is built to 1:24 scale. The real car is 4.8 metres long. How long is the model? If the real car's bonnet has area 2.4 m², what is the model bonnet's area? Why is it not simply 2.4 / 24?
0/5QUESTS
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A photocopier enlarges a photo so every dimension doubles. The original photo is 10 cm by 15 cm. What are the new dimensions? What happens to the area? Does the area also double? Explain your reasoning.
Record your answer in your workbook.
1
The Big Idea
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Similar figures have the same shape but not necessarily the same size. Every length is multiplied by the same number -- the scale factor. But areas scale by the square of the scale factor, and volumes scale by the cube. This is one of the most powerful ideas in geometry.
If the scale factor is $k$, then lengths become $k$ times longer, areas become $k^2$ times larger, and volumes become $k^3$ times larger. A scale factor greater than 1 is an enlargement. Less than 1 is a reduction.
Scale factorThe ratio of corresponding lengths in similar figures. $k = \frac{\text{new}}{\text{original}}$.
EnlargementA transformation with scale factor $k > 1$. The image is larger than the original.
ReductionA transformation with scale factor $0 < k < 1$. The image is smaller than the original.
CorrespondingMatching parts in the same relative position in similar figures.
ProportionalIn the same ratio. Corresponding sides of similar figures are proportional.
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Spot the Trap
heads-up
Wrong: Similar triangles must be the same size.
Right: Similar triangles have the same shape (equal angles) but can be different sizes. The ratio of corresponding sides is the scale factor.
Wrong: The scale factor is the difference between corresponding sides.
Right: The scale factor is the ratio of corresponding sides, not the difference. Scale factor = new length / original length.
5
The Three Similarity Tests
+5 XP
Unlike congruence which has four tests, similarity has three. The key difference: similarity only requires the same shape, not the same size. So sides can be in proportion rather than equal.
AAA (Angle-Angle-Angle): all corresponding angles equal. Since angles fix the shape, the sides must be in proportion. SSS ratio: all three pairs of sides in the same ratio. SAS ratio + angle: two sides in ratio with the included angle equal.
AAA | SSS ratio | SAS ratio + angle
AAA needs only two angles
The third angle is fixed by 180°. So two equal angles is enough.
SSS ratio = proportion
Check: $\frac{a}{d} = \frac{b}{e} = \frac{c}{f}$.
Congruent is similar
When $k = 1$, similar becomes congruent.
6
Scale Factor
+5 XP
The scale factor $k$ is the multiplier that takes every length in the original figure to the corresponding length in the similar figure. It is always a ratio, never a difference.
If $k > 1$, the image is an enlargement -- bigger than the original. If $0 < k < 1$, the image is a reduction -- smaller. If $k = 1$, the figures are congruent. The scale factor formula is always $k = \frac{\text{image length}}{\text{original length}}$.
$k = \frac{\text{image}}{\text{original}}$
Always divide new by old
$k = \frac{\text{new}}{\text{original}}$ never the other way around.
Check with a second pair
Verify $k$ is the same for all corresponding sides.
k = 1 means congruent
Congruent triangles are a special case of similar.
7
Area and Volume Scale Factors
+5 XP
When lengths scale by $k$, areas scale by $k^2$ and volumes scale by $k^3$. This is because area depends on two dimensions (length and width) and volume depends on three (length, width, and height).
Think of a square with side $s$. Its area is $s^2$. If the side becomes $ks$, the area becomes $(ks)^2 = k^2 s^2$ -- that is $k^2$ times the original area. For a cube, volume goes from $s^3$ to $(ks)^3 = k^3 s^3$ -- $k^3$ times the original.
$A_{\text{new}} = k^2 \cdot A_{\text{old}}$
Square for area
If $k = 3$, area is $3^2 = 9$ times larger.
Cube for volume
If $k = 3$, volume is $3^3 = 27$ times larger.
Work backwards too
From area ratio, $k = \sqrt{\text{area ratio}}$.
Watch Me Solve It · 3 examples
Watch Me Solve It · Find missing sides
+15 XP per step
Q1
PROBLEM
$\triangle ABC \sim \triangle DEF$. $AB = 6$ cm, $BC = 8$ cm, $AC = 10$ cm, and $DE = 9$ cm. Find $EF$ and $DF$.
1
Find the scale factor
$k = \frac{DE}{AB} = \frac{9}{6} = 1.5$
Scale factor = image length / original length. $DE$ corresponds to $AB$.
2
Apply the scale factor to the other sides
$EF = BC \times k = 8 \times 1.5 = 12$ cm
$DF = AC \times k = 10 \times 1.5 = 15$ cm
Every side in the image is 1.5 times the corresponding original side.
Divide the larger area by the smaller area to get the area ratio.
2
Find the linear scale factor
$k^2 = 4$ → $k = \sqrt{4} = 2$
Since area scales by $k^2$, the linear scale factor is the square root of the area ratio.
3
Find the unknown side
Shortest side of larger triangle = $4 \times 2 = 8$ cm
Multiply the original side by the linear scale factor $k = 2$.
Nice work -- XP earned
Answer$8$ cm
9
Common Pitfalls
heads-up
Using difference instead of ratio
Calculating scale factor as the difference between corresponding sides (e.g., $9 - 6 = 3$) instead of the ratio ($9 / 6 = 1.5$). Scale factor is always a multiplicative relationship, never additive.
Fix: always divide new by original. $k = \frac{\text{new}}{\text{original}}$.
Forgetting that area scales by $k^2$
When dimensions double, claiming the area also doubles. Area depends on two dimensions, so it quadruples. This error is extremely common in exam questions.
Fix: write "length $\times k$, area $\times k^2$, volume $\times k^3$" on every similar figure problem.
Mixing up corresponding sides
Pairing the shortest side of one triangle with the longest side of the other. This gives an inconsistent scale factor and incorrect answers for the remaining sides.
Fix: always match shortest to shortest, middle to middle, longest to longest. Verify $k$ is the same for all pairs.
Copy Into Your Books
Similarity Tests
AAA -- all angles equal
SSS ratio -- sides proportional
SAS -- two sides in ratio + included angle
Scale Factor
$k = \frac{\text{new}}{\text{original}}$
$k > 1$ = enlargement
$k < 1$ = reduction
$k = 1$ = congruent
Area and Volume
Length: $\times k$
Area: $\times k^2$
Volume: $\times k^3$
Working Backwards
From area ratio: $k = \sqrt{\text{ratio}}$
From volume ratio: $k = \sqrt[3]{\text{ratio}}$
How are you completing this lesson?
Brain Trainer · 4 problems
D
Brain Trainer · Scale factor drill
4 problems
Four quick problems on scale factors. Work each one, then reveal the answer.
1 $\triangle ABC \sim \triangle DEF$. $AB = 4$ cm, $DE = 12$ cm. Find the scale factor.
$EF = BC \times k = 6 \times 3 = 18$ cm$EF = 18$ cm
3 $\triangle ABC \sim \triangle DEF$. $AC = 9$ cm, $DF = 3$ cm. Find the scale factor and state whether it is an enlargement or reduction.
$k = \frac{DF}{AC} = \frac{3}{9} = \frac{1}{3}$. Since $k < 1$, this is a reduction.$k = \frac{1}{3}$, reduction
4 Two similar cubes have side lengths 2 cm and 6 cm. By what factor does the volume increase?
$k = \frac{6}{2} = 3$. Volume scale factor = $k^3 = 3^3 = 27$. Small volume = $8$ cm³, large volume = $216$ cm³. $216 / 8 = 27$.Volume increases by factor of 27
Complete in your workbook.
Multiple Choice · 5 questions
MC1
Definition of similar
+10 XP
Two figures are similar if:
Correct -- similar means same shape with proportional sides and equal angles.
Similar figures can have different areas and perimeters. Congruent figures are a special case where the scale factor is 1.
Explanation: Similar figures have equal corresponding angles and proportional corresponding sides. Same area or perimeter does not guarantee similarity.
MC2
Calculating scale factor
+10 XP
The scale factor from a 3 cm side to a 9 cm corresponding side is:
Correct -- $k = 9 / 3 = 3$.
Scale factor = new / original = $9 / 3 = 3$.
Explanation: $k = \frac{\text{new length}}{\text{original length}} = \frac{9}{3} = 3$. Since $k > 1$, this is an enlargement.
MC3
Area scale factor
+10 XP
If the scale factor is 2, the area scale factor is:
Correct -- area scales by $k^2 = 2^2 = 4$.
Area depends on two dimensions, so the area scale factor is $k^2$.
Explanation: If lengths are multiplied by 2, then area (length $\times$ width) is multiplied by $2 \times 2 = 4$. For example, a 3 by 4 rectangle has area 12. Doubled to 6 by 8, the area is 48. $48 / 12 = 4$.
MC4
Reduction or enlargement
+10 XP
A scale factor less than 1 produces:
Correct -- $k < 1$ means the image is smaller than the original: a reduction.
$k < 1$ makes every length smaller, so the figure is reduced.
Explanation: When $k < 1$, each length in the image is a fraction of the original length. For example, $k = 0.5$ means every side is halved -- a reduction. An enlargement requires $k > 1$.
MC5
Corresponding angles
+10 XP
Corresponding angles in similar triangles are:
Correct -- corresponding angles in similar triangles are always equal.
Similar triangles have the same shape, which means all corresponding angles are equal. It is the sides that are proportional, not the angles.
Explanation: The defining property of similar figures is that corresponding angles are equal and corresponding sides are in proportion. Angles determine shape; the scale factor determines size.
Short Answer · 3 questions
Q6
Similar triangle calculations
+15 XP
Q6
SHORT ANSWER
$\triangle ABC \sim \triangle DEF$. $AB = 5$ cm, $BC = 7$ cm, $AC = 9$ cm, and $DE = 15$ cm. (a) Find the scale factor. (b) Find the lengths of $EF$ and $DF$. (c) If the area of $\triangle ABC$ is 14 cm², what is the area of $\triangle DEF$?
Write your working in your book.
(a) $k = \frac{DE}{AB} = \frac{15}{5} = 3$.
(b) $EF = BC \times k = 7 \times 3 = 21$ cm. $DF = AC \times k = 9 \times 3 = 27$ cm.
(c) Area scale factor = $k^2 = 3^2 = 9$. Area of $\triangle DEF = 14 \times 9 = 126$ cm².
Marking guidance: 1 mark for (a), 1 mark for each side in (b), 1 mark for (c).
Q7
Photo enlargement
+15 XP
Q7
SHORT ANSWER
A photograph is enlarged so that its dimensions are tripled. By what factor does the area increase? Explain your reasoning.
Write your working in your book.
The linear scale factor is $k = 3$ (dimensions are tripled).
Area depends on two dimensions: length and width. Both are multiplied by 3, so the area is multiplied by $3 \times 3 = 9$.
Therefore the area increases by a factor of 9.
Example: original 10 cm by 15 cm has area 150 cm². Tripled: 30 cm by 45 cm has area 1350 cm². $1350 / 150 = 9$.
Marking guidance: 1 mark for identifying $k = 3$, 1 mark for explaining area scale factor $k^2$, 1 mark for final answer of 9.
Q8
Congruent vs similar
+15 XP
Q8
SHORT ANSWER
Explain the difference between congruent and similar triangles. Use diagrams to support your answer.
Write your working in your book.
Congruent triangles are identical in every way -- same shape AND same size. All corresponding sides are equal and all corresponding angles are equal. The scale factor is $k = 1$.
Similar triangles have the same shape but can be different sizes. All corresponding angles are equal, but corresponding sides are in proportion (not necessarily equal). The scale factor can be any positive number.
Diagram description: Draw two triangles with the same three angles (e.g., 50°, 60°, 70°). The congruent pair has sides 5, 6, 7 and 5, 6, 7. The similar pair has sides 5, 6, 7 and 10, 12, 14 (scale factor 2).
Marking guidance: 1 mark for correct definitions, 1 mark for explaining scale factor difference, 1 mark for diagram description.
S
Stretch Challenge · Model village
+20 XP
S
STRETCH
A model village is built to a scale of 1:50. (a) A real house is 12 m tall. How tall is the model in centimetres? (b) The model house has a roof area of 80 cm². What is the real roof area in square metres? (c) The real house requires 150 litres of paint for its exterior. How much paint would the model require? Explain why this might not be practical.
Record in your book -- full marks require clear working.
Reveal solution
(a) $k = \frac{1}{50}$. Model height = $12 \times \frac{1}{50} = 0.24$ m = 24 cm.
(b) Area scale factor = $k^2 = \frac{1}{2500}$. Real area = $80 \times 2500 = 200{,}000$ cm² = 20 m².
(c) Volume (paint) scale factor = $k^3 = \frac{1}{125{,}000}$. Model paint = $150 \times \frac{1}{125{,}000} = 0.0012$ litres = 1.2 mL. This is not practical because paint does not scale -- a minimum thickness is needed for coverage, and surface texture effects do not shrink proportionally.
R
Quick Review
Similar
Same shape, proportional sides
Scale factor
$k = \text{new} / \text{original}$
Enlargement
$k > 1$
Reduction
$0 < k < 1$
Area
Scales by $k^2$
Volume
Scales by $k^3$
Interactive: Circle Explorer
Explore circle geometry and properties in the interactive below. While this lesson focuses on similarity, understanding circles deepens your geometric reasoning.
Consolidation Game -- Doodle Jump Quiz
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Jump your way to the top by answering questions on similarity, scale factors, and area/volume scaling. The higher you climb, the harder the questions.
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