Ratio, proportion and rates of change

Scale factors, scale diagrams and maps

14 past-paper questions, easiest first. Try each one, then reveal the answer.

Foundation & Higher12 of 14 from Foundation papers

Revision guide by The Full Marks

Scale factors, scale diagrams and maps: what you need to know

Scale drawings and maps shrink real objects by a fixed amount so they fit on paper. You need to go both ways: from the drawing to real life and back, getting the units right.

Key facts

A scale of $1 : 25\,000$ means $1$ cm on the map is $25\,000$ cm in real life.
A scale like '1 cm to 4 m' already includes a unit change.
Real length $=$ map length $\times$ scale factor; map length $=$ real length $\div$ scale factor.
No scale given? Find it from a length you know: scale factor $=$ real length $\div$ drawing length.
Areas scale by the scale factor squared: with $1$ cm to $4$ m, $1\text{ cm}^2$ is $16\text{ m}^2$ .

Map or drawing to real life

Measure the length on the map carefully in cm.
Multiply by the scale number.
Convert to sensible units: divide by 100 for m, then by 1000 for km (or by $100\,000$ from cm to km).

Real life to drawing

Put the real length in the same unit as the scale.
Divide by the scale number.
Draw with a sharp pencil and ruler; drawn lengths are usually marked to within about 2 mm.

Worked examples

Example 1 Foundation · grade 3

A map has a scale of $1 : 25\,000$ . Two villages are $6.4$ cm apart on the map. How far apart are they in real life, in km?

Show the solution

$6.4 \times 25\,000 = 160\,000$ cm
$160\,000 \div 100 = 1600$ m
$1600 \div 1000 = 1.6$ km

Example 2 Foundation · grade 5

Ella makes a scale drawing of a rectangular garden using $1$ cm to $4$ m. The garden is $26$ m by $18$ m. (a) What size is the drawing? (b) The drawing's area is $29.25\text{ cm}^2$ . Use it to find the real area.

Show the solution

(a) $26 \div 4 = 6.5$ cm and $18 \div 4 = 4.5$ cm
(b) $1$ cm $^2$ represents $4 \times 4 = 16\text{ m}^2$
$29.25 \times 16 = 468\text{ m}^2$ (check: $26 \times 18 = 468$ )

Common mistakes

Multiplying when you should divide (or the other way round). Real things are bigger than drawings.
Not converting first: with a scale of $1 : 30$ , a real length of $3.6$ m must become $360$ cm before you divide, giving $12$ cm.
Dividing by 1000 to go from cm to km. It's $100\,000$ .
Scaling an area by the length factor instead of its square.
Measuring the wrong length on the drawing, e.g. a height when the known length is a width. Use the matching length to find the scale.

Exam tips

Write units with every answer; a map answer without km or m can lose the final mark.
If you have to measure, show the length you measured so the examiner can follow your method.

Questions people ask

What does a map scale of 1:50 000 mean?

Every 1 cm on the map is 50 000 cm, which is 500 m, in real life. So 2 cm on the map is 1 km.

How do you use a scale drawing to find a real length?

Measure the length on the drawing, then multiply by the scale factor and convert to sensible units.

Exam questions: Scale factors, scale diagrams and maps (14, easiest first)

Scale factors, scale diagrams and maps2 marksGrade 2No calculatorFoundation

November 2023 · Paper 1 Foundation · Q7

Use a map scale to change a drawn length to a real distance and back again.

Show answer

(a)	15 m	1 mark
(b)	4 cm	1 mark

Worked solution Checked against the mark scheme

(a) Each cm is 5 m, so $3 \times 5 = 15$ m.

(b) $20 \div 5 = 4$ cm.

Watch out: Multiplying in (b) to get 100 cm.

Scale factors, scale diagrams and maps3 marksGrade 2CalculatorFoundation

June 2023 · Paper 2 Foundation · Q8

Use a map scale to find the real distance of a journey made of two measured legs.

Show answer

Answer	96 km	3 marks

Worked solution Checked against the mark scheme

Total map distance: $8 + 16 = 24$ cm

Each cm is 4 km: $24 \times 4 = 96$ km

Watch out: Only converting one leg of the journey, e.g. $16 \times 4 = 64$ .

Scale factors, scale diagrams and maps2 marksGrade 2CalculatorFoundation

November 2021 · Paper 2 Foundation · Q5

Make a scale drawing of a rectangle using a given scale.

Show answer

Answer	A rectangle 7 cm by 4 cm drawn on the grid	2 marks

Worked solution Checked against the mark scheme

Each 5 m becomes 1 cm.

$35 \div 5 = 7$ cm and $20 \div 5 = 4$ cm.

Draw a rectangle 7 cm long and 4 cm wide.

Watch out: Multiplying by 5 instead of dividing.

Scale factors, scale diagrams and maps2 marksGrade 2CalculatorFoundation

November 2018 · Paper 2 Foundation · Q8

Use a map scale to find a real distance in kilometres.

Show answer

Answer	263.2 km	2 marks

Worked solution Checked against the mark scheme

Each 1 cm stands for 14 km.

$18.8 \times 14 = 263.2$ km

Watch out: Dividing 18.8 by 14.

Scale factors, scale diagrams and maps4 marksGrade 2CalculatorFoundation

June 2024 · Paper 3 Foundation · Q6

Use a map scale to find a real distance and a map length.

Show answer

(a)	$32$ km	2 marks
(b)	$2.5$ cm (or 25 mm)	2 marks

Worked solution Checked against the mark scheme

(a) Each 1 cm is 4 km, so $8 \times 4 = 32$ km.

(b) $10 \div 4 = 2.5$ , so the road is $2.5$ cm long on the map.

Watch out: Leaving off the units in (b), which loses the accuracy mark.

Scale factors, scale diagrams and maps3 marksGrade 3CalculatorFoundation

June 2022 · Paper 2 Foundation · Q16

Use a scale drawing to find the real width of a building from its real length.

Show answer

Answer	47 metres	3 marks

Worked solution Checked against the mark scheme

Scale: $62 \div 12.4 = 5$ , so 1 cm on the drawing is 5 m in real life.

Real width: $9.4 \times 5 = 47$ m.

Watch out: Mixing units, e.g. treating 62 m as 62 cm with the wrong scale factor.

Scale factors, scale diagrams and maps4 marksGrade 3CalculatorFoundation

November 2021 · Paper 3 Foundation · Q12

Measure a map distance and use a scale to find the real distance, then use a ratio scale to convert a drawing length to metres.

Show answer

(a)	50 km	2 marks
(b)	60 m	2 marks

Worked solution Checked against the mark scheme

(a) Measure the distance on the map: about 2.5 cm.

$2.5 \times 20 = 50$ km

(b) $5 \times 1200 = 6000$ cm on the ground.

$6000 \div 100 = 60$ m

Watch out: In (b), giving 6000 without converting centimetres to metres.

Scale factors, scale diagrams and maps2 marksGrade 3CalculatorFoundation

June 2019 · Paper 3 Foundation · Q17

Use a scale drawing to estimate the real height of a mast from a box of known width.

Show answer

Answer	9 (accept 8 to 10)	2 marks

Worked solution Checked against the mark scheme

Measure: the box is about 1.8 cm wide and the mast about 10.8 cm tall.

The mast is about $10.8 \div 1.8 = 6$ times the box width.

Height $\approx 6 \times 1.5 = 9$ m

Watch out: Measuring the box's height instead of its width.

Scale factors, scale diagrams and maps2 marksGrade 3CalculatorFoundation

Specimen · Paper 2 Foundation · Q12

Use a model scale to find a model's length in centimetres from a length in metres.

Show answer

Answer	12 cm	2 marks

Worked solution Checked against the mark scheme

$3.6$ m $= 360$ cm

$360 \div 30 = 12$ cm

Watch out: Dividing 3.6 by 30 and getting 0.12.

Scale factors, scale diagrams and maps3 marksGrade 4CalculatorFoundation

November 2024 · Paper 2 Foundation · Q15

Use a map scale to find a real length in kilometres.

Show answer

Answer	3.5	3 marks

Worked solution Checked against the mark scheme

Real length $= 14 \times 25\,000 = 350\,000$ cm.

There are $100\,000$ cm in 1 km, so $350\,000 \div 100\,000 = 3.5$ km.

Watch out: Dividing by 1000 as if converting metres to kilometres.

Scale factors, scale diagrams and maps5 marksGrade 4CalculatorFoundation

June 2018 · Paper 3 Foundation · Q12

Measure a scale drawing, use a 1 : 200 scale, and find the real perimeter in metres.

Show answer

Answer	$69.2$ m (accept 67.6 to 70.8)	5 marks

Worked solution Checked against the mark scheme

Measure the drawing: length about 11.5 cm, width about 5.8 cm.

Real length: $11.5 \times 200 = 2300$ cm $= 23$ m.

Real width: $5.8 \times 200 = 1160$ cm $= 11.6$ m.

Perimeter: $2 \times 23 + 2 \times 11.6 = 69.2$ m.

Watch out: Forgetting to convert from centimetres to metres.

Scale factors, scale diagrams and maps3 marksGrade 4No calculatorFoundation

Specimen · Paper 1 Foundation · Q15

Use a scale plan to find the real size of a floor and decide whether 150 boards are enough to cover it.

Show answer

Answer	No: the floor is 24 m by 8 m, so 160 boards are needed	3 marks

Worked solution Checked against the mark scheme

Measure the plan: about 12 cm by 4 cm.

Scale 1 cm : 2 m, so the floor is $24$ m by $8$ m.

Area $= 24 \times 8 = 192$ m². Each board covers $1.2 \times 1 = 1.2$ m².

$192 \div 1.2 = 160$ boards are needed ( $20$ along $\times$ $8$ across), which is more than 150. So she does not have enough.

Watch out: Forgetting to apply the scale to both lengths, or dividing by the scale instead of multiplying.

Scale factors, scale diagrams and maps3 marksGrade 6CalculatorHigher

June 2024 · Paper 2 Higher · Q8

Use a scale drawing's area and one real side of a right-angled triangle to find another real side.

Show answer

Answer	$35$ m	3 marks

Worked solution Checked against the mark scheme

On the drawing, $QR = 40 \div 5 = 8$ cm.

Area $= \frac12 \times PQ \times 8 = 28$ , so $PQ = 7$ cm on the drawing.

Real length $= 7 \times 5 = 35$ m.

Watch out: Scaling the area by 5 instead of $5^2 = 25$ when converting between real and drawing areas.

Scale factors, scale diagrams and maps3 marksGrade 6CalculatorHigher

November 2023 · Paper 3 Higher · Q11

Use a scale to convert an area on a plan into a real area in square metres.

Show answer

Answer	$12$	3 marks

Worked solution Checked against the mark scheme

Length scale factor 50, so area scale factor $50^2 = 2500$ .

Real area $= 48 \times 2500 = 120\,000\text{ cm}^2$

$1\text{ m}^2 = 10\,000\text{ cm}^2$ , so $120\,000 \div 10\,000 = 12\text{ m}^2$

Watch out: Multiplying by 50 instead of 50², or dividing by 100 instead of 10 000 to convert to square metres.

Also tested in these questions

Their main topic is different, but part of each one tests this topic.

June 2023 · Paper 3 Foundation · Q17: Plot a point from a bearing and a distance using a map scale.
November 2022 · Paper 1 Foundation · Q14: Find a distance from a speed and a time in hours and a half, then convert a map length to real kilometres using a scale.
June 2022 · Paper 3 Foundation · Q13: Measure distances on a scale map to compare two routes, then measure a bearing.
November 2020 · Paper 1 Foundation · Q13: Measure a bearing between two points on an accurate map, then use a map ratio to find the real distance in kilometres.
November 2019 · Paper 1 Foundation · Q27: On a scale drawing, plot a boat's position from a bearing, speed and time, then measure its distance and bearing from a lighthouse.
June 2019 · Paper 2 Foundation · Q19: Shade the region on a scale plan of a kitchen that is more than a set distance from a corner and from a wall.