MathsICSE Class 10 · Similarity

Maps and Models: Scale Factor for Length, Area and Volume (ICSE Class 10)

Updated 15 October 2026 · Fact-checked

A map or model drawn to the scale 1 : n is similar to the real object with scale factor k = 1/n. Real lengths are n times map lengths, real areas are n² times map areas, and real volumes are n³ times model volumes. Convert units carefully: 1 m² = 10,000 cm² and 1 km² = 10¹⁰ cm².

Going from map to real, lengths multiply by n, areas by n², volumes by n³; convert units before you multiply.

Illustration drawn for QuestionBuddy.

Understand Maps and Models (Scale Factor)

The CISCE syllabus lists maps and models as an application of similarity. A map or a model is a similar figure to the real thing.

Scale. A scale of $1 : n$ means 1 unit on the map or model stands for $n$ of the same units in reality. The scale factor from real to map is $k = \frac{1}{n}$ .

Three rules.

Length: real length $= n \times$ map length.
Area: real area $= n^{2} \times$ map area.
Volume: real volume $= n^{3} \times$ model volume. (Capacity and, for the same material, mass also scale by $n^{3}$ .)

Each rule follows from similarity: lengths scale by $n$ ; an area is a length times a length; a volume is length × length × length.

Units. Work in centimetres, then convert.

$1 m = 100 cm$ , $1 km = 100000 cm$ .
$1 m^{2} = 1 0^{4} cm^{2}$ , $1 km^{2} = 1 0^{10} cm^{2}$ , $1 hectare = 1 0^{4} m^{2}$ .
$1 m^{3} = 1 0^{6} cm^{3}$ , $1 litre = 1000 cm^{3}$ .

The 2025 report on Q2(iii) lists errors converting $576 m^{2}$ to $cm^{2}$ and forgetting to convert at all.

Finding the scale. If a map area and the real area are known, $n^{2} = \frac{real area}{map area}$ (same units), so $n$ is the square root. With volumes, take the cube root.

Model to real or real to model? Real is bigger for maps and most models, so multiply when going to reality and divide when going to the model. A quick size check catches most slips: a real field is not 3 cm long.

Maps and Models (Scale Factor): formulas to remember

Scale factor: $scale 1 : n \Rightarrow k = \frac{1}{n}$; Map or model to reality.
Lengths: $real length = n \times map length$; Same units on both sides.
Areas: $real area = n^{2} \times map area$; Square the scale.
Volumes: $real volume = n^{3} \times model volume$; Cube the scale.
Area conversions: $1 m^{2} = 1 0^{4} cm^{2}, 1 km^{2} = 1 0^{10} cm^{2}$; Square the length conversion.

How to solve Maps and Models (Scale Factor) questions

Decide whether it is length, area or volume; use n, n² or n³; then convert units.

1Write the scale as $1 : n$ .
2Identify the quantity: length (use $n$ ), area (use $n^{2}$ ) or volume (use $n^{3}$ ).
3Multiply to go from map/model to reality; divide to go the other way.
4Work in cm, cm² or cm³ first, then convert to m, km, m², km², m³ or litres.
5To find the scale, divide real by map (same units), then take the square or cube root if needed.
6Check the size of the answer is sensible.

Maps and Models (Scale Factor) in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Map distance to real distance

Suppose a hiking map has scale $1 : 50000$ and a trail measures 9.6 cm on it.

How long is the trail, and how long will it take at 4 km/h?

Try it first, then show the solutionSolution

Real length $= 9.6 \times 50000 = 480000$ cm $= 4.8$ km.
Time $= \frac{4.8}{4} = 1.2$ h $= 72$ minutes.

Answer: 4.8 km; 72 minutes.

Memory hook: Lengths: multiply by n, then convert cm to km by ÷ 100 000.

Model lengths and areas

Imagine a doll's house built to the scale $1 : 12$ . A real door is 2.1 m tall and a real room has floor area 18 m².

Find the height of the model door and the floor area of the model room.

Try it first, then show the solutionSolution

Door: $\frac{210}{12} = 17.5$ cm.
Floor: $\frac{18}{1 2 ^{2}} = \frac{18}{144} = 0.125$ m² $= 1250$ cm².

Answer: 17.5 cm; 1250 cm².

Memory hook: Going down to the model, divide: by n for lengths, by n² for areas.

Tricks to remember Maps and Models (Scale Factor)

Short memory aids. Say each one aloud once, then use it on a question the same day.

1, 2, 3 powers
Length uses $n^{1}$ , area $n^{2}$ , volume $n^{3}$ : the power matches the dimension.
Use it when: Choosing the factor.
Square the conversion too
100 cm in a metre means $10 0^{2}$ cm² in a square metre and $10 0^{3}$ cm³ in a cubic metre.
Use it when: Unit conversions, the 2025 trouble spot.
Big world, small map
Real answers are bigger: multiply going out to reality, divide coming in to the map.
Use it when: Checking direction.

Maps and Models (Scale Factor): solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: A map distance · 1 mark · MCQBoard level

On a map with scale $1 : 25000$ , two towers are 8 cm apart. The actual distance is:
(a) 200 m
(b) 2 km
(c) 20 km
(d) 0.2 km

Try it first, then show the solutionSolution

$8 \times 25000 = 200000$ cm $= 2$ km.

Answer: (b) 2 km

Example 2: Distance and area on a map · 3 marks · Short answerBoard level

A map is drawn to the scale $1 : 40000$ .
(a) Find the actual distance for 3.5 cm on the map.
(b) A lake covers 2.5 cm² on the map. Find its actual area in km².

Try it first, then show the solutionSolution

(a) $3.5 \times 40000 = 140000$ cm $= 1.4$ km.
(b) Area factor $4000 0^{2} = 1.6 \times 1 0^{9}$ . Real area $= 2.5 \times 1.6 \times 1 0^{9} = 4 \times 1 0^{9}$ cm².
$\frac{4 \times 1 0 ^{9}}{1 0 ^{10}} = 0.4$ km².

Answer: (a) 1.4 km (b) 0.4 km²

Example 3: A model ship · 3 marks · Short answerHarder board level

A model of a ship is made to the scale $1 : 150$ .
(a) The model is 40 cm long. Find the length of the ship in metres.
(b) The model's deck has area 300 cm². Find the area of the real deck in m².

Try it first, then show the solutionSolution

(a) $40 \times 150 = 6000$ cm $= 60$ m.
(b) $300 \times 15 0^{2} = 300 \times 22500 = 6750000$ cm² $= 675$ m².

Answer: (a) 60 m (b) 675 m²

Example 4: Finding the scale from an area · 4 marks · Short answerHarder board level

A field of area 1.44 km² is shown on a map as 4 cm².
(a) Find the scale of the map.
(b) Two points are 5 cm apart on this map. How far apart are they in reality?

Try it first, then show the solutionSolution

(a) $1.44 km^{2} = 1.44 \times 1 0^{10} cm^{2}$ . $n^{2} = \frac{1.44 \times 1 0 ^{10}}{4} = 3.6 \times 1 0^{9}$ .
$n = 3.6 \times 1 0^{9} = 60000$ . Scale $1 : 60000$ .
(b) $5 \times 60000 = 300000$ cm $= 3$ km.

Answer: (a) $1 : 60000$ (b) 3 km

Example 5: Volume and surface of a model tank · 4 marks · Short answerHOTS / competency

A model of a water tank is made to the scale $1 : 40$ . The model holds 0.5 litres, and its outer surface has area 150 cm².
(a) Find the capacity of the real tank in m³.
(b) Find the area of the real tank's surface in m².

Try it first, then show the solutionSolution

(a) Volume factor $4 0^{3} = 64000$ : $0.5 \times 64000 = 32000$ litres $= 32$ m³ (1 m³ = 1000 litres).
(b) Area factor $4 0^{2} = 1600$ : $150 \times 1600 = 240000$ cm² $= 24$ m².

Answer: (a) 32 m³ (b) 24 m²

Example 6: Case study: a park on a town map · 5 marks · Case-basedHOTS / competency

A town map has scale $1 : 5000$ . A rectangular park measures 6 cm by 4 cm on the map, and a triangular pond inside it has base 2 cm and height 1.5 cm on the map.
(a) Find the real length and breadth of the park.
(b) Find the real area of the park in m² and in hectares.
(c) Find the real area of the pond in m².

Try it first, then show the solutionSolution

(a) $6 \times 5000 = 30000$ cm $= 300$ m; $4 \times 5000 = 200$ m.
(b) $300 \times 200 = 60000$ m² $= 6$ hectares.
(c) Map area $= \frac{1}{2} \times 2 \times 1.5 = 1.5$ cm². Real area $= 1.5 \times 500 0^{2} = 37500000$ cm² $= 3750$ m².

Answer: (a) 300 m by 200 m (b) 60 000 m² = 6 ha (c) 3750 m²

Where students lose marks in Maps and Models (Scale Factor)

The slips examiners see most often, and what to write instead.

Mistake: Using n instead of n² for areas
Area scales by the square of the length scale.
Wrong working
Scale $1 : 150$ ; deck $300 cm^{2} \to 300 \times 150 = 45000$ cm².
Right working
$300 \times 15 0^{2} = 6750000$ cm².
Do this: Ask: is this a length, an area or a volume?
Usually costs: The final answer.
Mistake: Wrong area-unit conversion
$1 m^{2} = 1 0^{4} cm^{2}$ , not $1 0^{2}$ .
Wrong working
$576 m^{2} = 57600 cm^{2}$ .
Right working
$576 m^{2} = 5760000 cm^{2}$ .
Do this: Square the length conversion.
Usually costs: Reported in 2025 Q2(iii).
Mistake: Taking k² as the scale
The scale is the length ratio $k$ ; the area ratio is $k^{2}$ .
Wrong working
Area ratio $\frac{1}{40 000}$ , so the scale is $1 : 40000$ .
Right working
$k^{2} = \frac{1}{40 000}$ , $k = \frac{1}{200}$ : scale $1 : 200$ .
Do this: Take the square root of an area ratio to get the scale.
Usually costs: Reported in 2025 Q2(iii).

Exam tips

2025 Q2(iii) (4 marks) found an area of 144 cm² and compared it with 576 m² to get $k^{2}$ , then the scale $1 : 200$ .
Write the factor you are using ('area factor $= n^{2} = \dots$ ') before multiplying.
Give answers in the units asked: km for distances on maps, m² or hectares for land, litres or m³ for capacity.
Use powers of ten to keep large numbers manageable.

Practice questions on Maps and Models (Scale Factor)

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
A model of a statue is made to the scale $1 : 50$ . The model is 12 cm tall. The statue's height is:
(a) 60 cm
(b) 6 m
(c) 60 m
(d) 0.6 m
Hint
$12 \times 50$ cm.
Answer
(b) 6 m
Q2 · 2 marks · Short answerBoard level
Two towns are 15 km apart. How far apart are they on a map of scale $1 : 200000$ ?
Hint
$15 km = 1500000$ cm.
Answer
7.5 cm
Q3 · 3 marks · Short answerHarder board level
On a map of scale $1 : 25000$ , a forest covers 12 cm². Find its real area in km².
Hint
Area factor $2500 0^{2} = 6.25 \times 1 0^{8}$ .
Answer
$12 \times 6.25 \times 1 0^{8} = 7.5 \times 1 0^{9}$ cm² $= 0.75$ km²
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): On a map of scale $1 : 1000$ , an area of 1 cm² represents 1000 cm².
Reason (R): Areas on a map of scale $1 : n$ are multiplied by $n^{2}$ to give real areas.
(a) (A) is true and (R) is false.
(b) (A) is false and (R) is true.
(c) Both (A) and (R) are true, and (R) is the correct explanation for (A).
(d) Both (A) and (R) are true, but (R) is not the correct explanation for (A).
Hint
$100 0^{2} = 1 0^{6}$ .
Answer
(b). 1 cm² represents $1 0^{6}$ cm² = 100 m².
Q5 · 3 marks · Short answerHarder board level
A model aeroplane is made to the scale $1 : 80$ . (a) Its wingspan is 45 cm; find the real wingspan. (b) A window on the model has area 0.5 cm²; find the real window's area in m².
Hint
Use 80 for length and $8 0^{2} = 6400$ for area.
Answer
(a) 36 m (b) $0.5 \times 6400 = 3200$ cm² $= 0.32$ m²
Q6 · 3 marks · Short answerHarder board level
A clay model of a statue is made to the scale $1 : 20$ and uses 0.6 kg of clay. How much clay would a full-size statue of the same clay need?
Hint
Mass of the same material scales like volume.
Answer
$0.6 \times 2 0^{3} = 0.6 \times 8000 = 4800$ kg
Q7 · 3 marks · Short answerHOTS / competency
A lake covers 15 cm² on a map of scale $1 : 10000$ . What area will it cover on a map of scale $1 : 25000$ ?
Hint
Lengths on the second map are $\frac{10 000}{25 000} = 0.4$ times those on the first.
Answer
$15 \times 0. 4^{2} = 2.4$ cm²

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Scale $1 : n$ . Area factor?Show answer
$n^{2}$
2.Scale $1 : n$ . Volume factor?Show answer
$n^{3}$
3.How many cm² in 1 m²?Show answer
10 000
4.Map scale $1 : 100000$ : what does 1 cm represent?Show answer
1 km
5.Model $1 : 10$ holds 2 litres. Real capacity?Show answer
2000 litres

Maps and Models (Scale Factor): frequently asked questions

How do I find the actual area from a map?

Multiply the map area by n², where the scale is 1 : n, then convert units. For example, at 1 : 10 000, 1 cm² represents 10⁸ cm² = 10 000 m².

How does volume change for a model?

For scale 1 : n, real volume = n³ × model volume. A model at 1 : 10 that holds 1 litre represents 1000 litres.

How do I find the scale of a map from an area?

Divide the real area by the map area in the same units to get n², then take the square root to get n. The scale is 1 : n.