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Last update: 2025-08-20

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Crash report

Maps & plans

2025-08-20

Crash report

Unit 1: Angles & Constructions
Unit 2: Shapes & Symmetry
Unit 3: Position & Transformation
Unit 4: Area, Volume & Symmetry

🎯 In this topic you will

Use scales on maps and plans

🧠 Key Words

scale drawing
scale

Show Definitions

scale drawing: A drawing that shows an object or space in accurate proportion to its real size, reduced or enlarged using a consistent ratio.
scale: The ratio between a measurement on a drawing or map and the actual measurement in real life.

🖼️ What is a scale drawing?

A scale drawing is a drawing that represents an object in real life.

📏 What is a scale?

The scale gives the relationship between the lengths on the drawing and the real-life lengths.

✍️ Ways to write a scale

Using the word represents; for example, $1\ \text{cm}$ represents $100\ \text{cm}$.
Using the word to; for example, $1\ \text{to}\ 100$.
Using a ratio sign; for example, $1:100$.

⚠️ Same-units rule

When you write a scale using to or the ratio sign, the numbers must be in the same units.

🔁 Convert a verbal scale to a ratio

“$1\ \text{cm}$ represents $10\ \text{m}$” must be written as either $1\ \text{to}\ 1000$ or $1:1000$ (convert $10\ \text{m}=1000\ \text{cm}$).

🔎 What does $1:10$ mean?

The scale $1\ \text{to}\ 10$ or $1:10$ means every centimetre on the drawing represents $10$ centimetres in real life.

🔧 Using a scale

To change a length on a drawing to a real-life length, multiply by the scale.
To change a real-life length to a drawing length, divide by the scale.

📘 Worked example

a. Razaan makes a scale drawing of the front of a building. She uses a scale of 1 cm represents 5 m.

i. On her drawing, the building is 12 cm long. How long is the building in real life?

ii. The building in real life is 120 m tall. How tall is the building on the scale drawing?

b. A map has a scale of $1:50{,}000$.

i. On the map the distance between two villages is 8 cm. What is the distance between the two villages in real life?

ii. The distance between two towns is 18 km in real life. What is the distance between the two towns on the map?

Answer:

a. i. $12 \times 5 = 60\text{ m}$

Multiply the length on the drawing by the scale of 5 to get the real-life length. Remember to include the units (m) with your answer.

a. ii. $120 \div 5 = 24\text{ cm}$

Divide the height in real life by the scale of 5 to get the height of the drawing. Remember to include the units (cm) with your answer.

b. i. $8\text{ cm} \times 50{,}000 = 400{,}000\text{ cm}$

$400{,}000 \div 100 = 4000\text{ m}$

$4000 \div 1000 = 4\text{ km}$

The two villages are 4 km apart in real life.

b. ii. $18\text{ km} \times 1000 = 18{,}000\text{ m}$

$18{,}000 \times 100 = 1{,}800{,}000\text{ cm}$

$1{,}800{,}000 \div 50{,}000 = 36\text{ cm}$

The distance between the two towns on the map is 36 cm.

Multiply the length on the drawing by the scale to find the real-life size, or divide the real-life size by the scale to find the drawing size.

When converting map distances, always ensure units match (cm → m → km, or km → m → cm) before applying the scale factor.

🧠 PROBLEM-SOLVING Strategy

Scale Drawings & Map Scales

Convert between drawing length and real distance using a single scale factor and consistent units.

Read the scale and write it as a ratio for one drawing unit: e.g., $1\ \text{cm} : 3\ \text{m}$ becomes $1\ \text{cm} : 300\ \text{cm}$ so the scale factor is $1:300$. A map scale $1\!:\!25{,}000$ means $1\ \text{cm}$ on the map represents $25{,}000\ \text{cm}$ ($250\ \text{m}$ or $0.25\ \text{km}$).
Make all quantities the same unit (usually centimetres):
- $1\ \text{m}=100\ \text{cm}$
- $1\ \text{km}=1000\ \text{m}=100{,}000\ \text{cm}$
Choose the operation:
- Drawing → Real:$\text{Real}=\text{Drawing}\times \text{scale\_factor}$
- Real → Drawing:$\text{Drawing}=\text{Real}\div \text{scale\_factor}$
Here, $\text{scale\_factor}$ is the real length represented by $1$ drawing unit in the same unit (e.g., centimetres).
Present the answer in requested units (e.g., convert cm → m or km) and include units clearly. Example conversion: $3000\ \text{cm}=30\ \text{m}$, $3000\ \text{m}=3\ \text{km}$.
Quick reasonableness check: a larger real distance should correspond to a larger scale factor $n$ in $1:n$; map lengths should be much smaller than real distances.
Finding the scale from data (multiple-choice like $1:65{,}000$, $1:70{,}000$, $1:75{,}000$):
1. Convert real distance to centimetres: e.g., $18\ \text{km}=1{,}800{,}000\ \text{cm}$.
2. Compute $n=\dfrac{\text{Real (cm)}}{\text{Drawing (cm)}}$. With $24\ \text{cm}$ on the map: $n=\dfrac{1{,}800{,}000}{24}=75{,}000$.
3. Write the scale as $1:n$ → $1:75{,}000$.

Conversion	Example
$\text{map }1:n \Rightarrow 1\ \text{cm} \mapsto n\ \text{cm}$	$1:80{,}000 \Rightarrow 1\ \text{cm} \mapsto 80{,}000\ \text{cm}=800\ \text{m}$
$\text{Real}=\text{Drawing}\times n$	$18\ \text{cm}\times 3=54\ \text{m}$
$\text{Drawing}=\text{Real}\div n$	$12\ \text{m}\div 4=3\ \text{cm}$

❓ EXERCISES

1. A scale drawing uses a scale of $1 \text{ cm}$ represents $3 \text{ m}$.
Copy and complete the workings.

a. $2 \text{ cm}$ on the drawing represents $2 \times 3 = \_\_\_\_ \text{ m}$ in real life.

b. $5 \text{ cm}$ on the drawing represents $5 \times 3 = \_\_\_\_ \text{ m}$ in real life.

c. $8 \text{ cm}$ on the drawing represents $\_\_\_ \times 3 = \_\_\_\_ \text{ m}$ in real life.

👀 Show answer

a. $2 \times 3 = 6 \text{ m}$
b. $5 \times 3 = 15 \text{ m}$
c. $8 \div 3 = 2.67 \text{ cm}$ (to get back cm) or $8 \times 3 = 24 \text{ m}$ in real life

2. A scale drawing uses a scale of $1 \text{ cm}$ represents $4 \text{ m}$.
Copy and complete the workings.

a. $8 \text{ m}$ in real life represents $8 \div 4 = \_\_\_\_ \text{ cm}$ on the drawing.

b. $12 \text{ m}$ in real life represents $12 \div 4 = \_\_\_\_ \text{ cm}$ on the drawing.

c. $20 \text{ m}$ in real life represents $\_\_\_ \div 4 = \_\_\_\_ \text{ cm}$ on the drawing.

👀 Show answer

a. $8 \div 4 = 2 \text{ cm}$
b. $12 \div 4 = 3 \text{ cm}$
c. $20 \div 4 = 5 \text{ cm}$

3. Marsile draws a scale drawing of a playing field. He uses a scale of $1 \text{ cm}$ represents $10 \text{ m}$.

a. On his drawing the playing field is $18 \text{ cm}$ long. How long is the playing field in real life?

b. The playing field in real life is $80 \text{ m}$ wide. How wide is the playing field on the scale drawing?

👀 Show answer

a. $18 \times 10 = 180 \text{ m}$
b. $80 \div 10 = 8 \text{ cm}$

❓ EXERCISES

4. The map shows part of Zimbabwe, in Africa. The scale of the map is $1\ \text{cm}$ represents $40\ \text{km}$.

a. Use a ruler to measure the distance, in cm, from Gweru to Masvingo. Write this down.

b. Use your measurement from part a to work out the distance, in km, from Gweru to Masvingo in real life.

👀 Show answer

a. The answer depends on the printed size of your map. Let your measured distance be $x$ cm (measured with a ruler on your copy).

b. Using the scale $1\ \text{cm} : 40\ \text{km}$, the real-life distance is

$\text{distance} = x \times 40\ \text{km}$.

Example: if you measured $x=3.0$ cm, then the distance is $3.0 \times 40 = 120\ \text{km}$.

Note: Measuring on a resized screen image will give a different $x$; use the printed page or a scale bar if provided.

🧠 Think like a Mathematician

Task: Consider the scale $1:20$. Marcus, Arun, and Sofia each give their own interpretation of what this means:

Marcus: "The scale 1:20 means that 1 cm on the scale drawing represents 20 m in real life."
Arun: "The scale 1:20 means that 1 mm on the scale drawing represents 20 cm in real life."
Sofia: "The scale 1:20 means that 1 cm on the scale drawing represents 20 cm in real life."

Question: Who is correct, Marcus, Arun, or Sofia? Explain your answer.

👀 show answer

Correct answer: Sofia is correct.
Reason: A scale of $1:20$ means 1 unit on the drawing corresponds to 20 units in real life, using the same units. So, 1 cm on the drawing equals 20 cm in real life.
Marcus's mistake: He incorrectly scaled up to metres (1 cm → 20 m).
Arun's mistake: He used millimetres but mis-scaled (1 mm → 20 cm would mean 1 cm → 200 cm, not 20 cm).

❓ EXERCISES

6. This scale drawing of Visuri’s bedroom is drawn on centimetre squared paper. The scale is 1 to 25.

a. Work out the length in real life of the wall:

i. $AB$ ii. $BC$ iii. $CD$ iv. $DE$ v. $EF$ vi. $AF$

Give your answers in metres.

b. The wardrobe in Visuri’s room is $50 \text{ cm}$ deep. What is this measurement on the scale drawing?

c. The bed in Visuri’s room is $1.75 \text{ m}$ long. What is this measurement on the scale drawing? Give your answer in centimetres.

👀 Show answer

a. Scale $1:25$ means $1 \text{ cm}$ on the drawing = $25 \text{ cm}$ in reality.

$AB = 6 \text{ cm} \times 25 = 150 \text{ cm} = 1.5 \text{ m}$
$BC = 4 \text{ cm} \times 25 = 100 \text{ cm} = 1.0 \text{ m}$
$CD = 5 \text{ cm} \times 25 = 125 \text{ cm} = 1.25 \text{ m}$
$DE = 4 \text{ cm} \times 25 = 100 \text{ cm} = 1.0 \text{ m}$
$EF = 2 \text{ cm} \times 25 = 50 \text{ cm} = 0.5 \text{ m}$
$AF = 7 \text{ cm} \times 25 = 175 \text{ cm} = 1.75 \text{ m}$

b. $50 \text{ cm} \div 25 = 2 \text{ cm}$

c. $1.75 \text{ m} = 175 \text{ cm}$ in real life. $175 \div 25 = 7 \text{ cm}$ on the drawing.

7. A map has a scale of $1:25{,}000$.

a. On the map the distance between two villages is $12 \text{ cm}$. What is the distance, in km, between the two villages in real life?

b. The distance between two schools is $12 \text{ km}$ in real life. What is the distance, in cm, between the two schools on the map?

👀 Show answer

a. $12 \times 25{,}000 = 300{,}000 \text{ cm} = 3{,}000 \text{ m} = 3 \text{ km}$

b. $12 \text{ km} = 1{,}200{,}000 \text{ cm}$. On map: $1{,}200{,}000 \div 25{,}000 = 48 \text{ cm}$

8. Aika and Hinata use different methods to answer Question 2a. This is what they write:

a. Critique Aika’s and Hinata’s methods. What are the advantages and disadvantages of each method?

b. Whose method do you prefer? Explain why.

👀 Show answer

a. Aika converts everything to cm, then to km. This is methodical but involves extra steps. Hinata converts step by step into larger units sooner, which is quicker but requires careful unit conversion.

b. Hinata’s method is clearer and more efficient, though Aika’s method reduces chances of error if you are unsure with unit conversions.

9. This map has a scale of $1:80{,}000$.

a. Use a ruler to measure the distance, in cm, from Letterston to Wolf’s Castle. Write this down.

b. Use your measurement from part a to work out the distance, in km, from Letterston to Wolf’s Castle in real life.

c. The distance from Wolf’s Castle to Fishguard is $12 \text{ km}$ in real life. What is this measurement, in cm, on the map?

👀 Show answer

a. (Student measures directly; suppose $\approx 2.5 \text{ cm}$)

b. $2.5 \times 80{,}000 = 200{,}000 \text{ cm} = 2{,}000 \text{ m} = 2 \text{ km}$

c. $12 \text{ km} = 1{,}200{,}000 \text{ cm}$. On map: $1{,}200{,}000 \div 80{,}000 = 15 \text{ cm}$

❓ EXERCISES

10. This is part of Faisal’s homework:

a. Explain the mistakes that Faisal has made.

b. Write the correct solution.

👀 Show answer

a. Faisal’s mistake is that he divided instead of multiplying. A scale of $1:50{,}000$ means $1 \text{ cm}$ on the map represents $50{,}000 \text{ cm}$ in real life. He should have multiplied $8.5 \text{ cm}$ by $50{,}000$, but instead he divided and then converted incorrectly.

b. Correct solution:

$8.5 \times 50{,}000 = 425{,}000 \text{ cm}$

$425{,}000 \div 100 = 4{,}250 \text{ m}$

$4{,}250 \div 1000 = 4.25 \text{ km}$

The train stations are $4.25 \text{ km}$ apart in real life.

❓ EXERCISES

11. Babra takes part in an $18\ \text{km}$ run. The distance of the route on a map is $24\ \text{cm}$. Work out if A, B or C is the correct map scale. Show your working.

A: $1:65000$ B: $1:70000$ C: $1:75000$

👀 Show answer

Actual distance $= 18\ \text{km} = 1\,800\,000\ \text{cm}$.
Scale $= \dfrac{\text{map distance}}{\text{actual distance}} = \dfrac{24}{1\,800\,000} = 1:75\,000$.

✅ Correct map scale: C ($1:75\,000$).

📘 What we've learned

Scales express the ratio between drawing length and real distance, e.g. $1:25{,}000$ means $1\ \text{cm}$ on the map represents $25{,}000\ \text{cm} = 250\ \text{m}$ in real life.
To find a real distance: $\text{Real} = \text{Drawing} \times \text{scale\_factor}$.
To find a drawing length: $\text{Drawing} = \text{Real} \div \text{scale\_factor}$.
All distances must be in the same unit before calculation (e.g., convert km → m → cm).
Map scale can also be calculated directly as $n = \dfrac{\text{Real (cm)}}{\text{Drawing (cm)}}$, giving a ratio $1:n$.
We practiced solving problems with everyday examples, checking that map lengths are much smaller than real distances for reasonableness.