Teach Me Maths

Scale Drawings and Maps

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How to use map scales written in words or as a ratio, change between map distances and real distances in different units, read scale drawings and plans of rooms, and work out a scale from two lengths.

Scales in the exam

A scale tells you how a length on a map or a drawing compares with the real length. Maps, plans of rooms, drawings of gardens and models of cars or boats are all drawn to a scale, so that something large fits on a page.

A scale is written in one of two ways. In words, such as "1 cm represents 5 km". Or as a ratio, such as 1:25\,000

In the exam you are usually asked to work out a real distance from a map, or a length on the map from a real distance, for 1 to 3 marks. Many of these questions also ask for the answer in a different unit, such as kilometres. Others give a scale drawing of a room or a field and ask for a real length, a perimeter or an area.

If the question says the drawing is accurate, measure the length you need with a ruler, to the nearest millimetre. Then use the scale.

Scales written in words

A scale such as "1 cm represents 4 km" means that every centimetre on the map stands for 4 km in real life. So 2 cm stands for 8 km, 3 cm for 12 km, and so on.

To go from the map to real life, multiply by the scale. To go from real life to the map, divide by the scale.

\text{real distance} = \text{map distance} \times 4

\text{map distance} = \text{real distance} \div 4

Check that your answer makes sense. A real distance is much bigger than the map distance, and a map distance is small enough to fit on the page.

Worked example 1: From the map to real life

On a map, 1 cm represents 3 km. Two harbours are 5.5 cm apart on the map. Work out the real distance between the harbours.

  1. Each centimetre on the map stands for 3 km, and there are 5.5 centimetres.
  2. Going from the map to real life, so multiply by 3

  3. Your turn. Work out 5.5 \times 3

    5.5 \times 3 = 16.5

  4. Check: 16.5 km is much more than 5.5 cm, as a real distance should be.
  5. Answer: 16.5 km

Worked example 2: From real life to the drawing

Amir is making a scale drawing of a sports hall. He uses a scale of 1 cm to represent 6 m. The real hall is 45 m long. How long is the hall on Amir's drawing?

  1. Going from real life to the drawing, so divide by 6
  2. Think of it as: how many lots of 6 m fit into 45 m? Each lot is 1 cm on the drawing.

  3. Your turn. Work out 45 \div 6

    45 \div 6 = 7.5

  4. Check: 7.5 \times 6 = 45, so 7.5 cm on the drawing stands for 45 m.
  5. Answer: 7.5 cm

Scales written as a ratio

A scale of 1:25\,000 means that 1 unit on the map stands for 25\,000 of the same unit in real life. Map lengths are measured in centimetres, so 1 cm on the map stands for 25\,000 cm in real life.

You use it in the same way. Multiply a map length by 25\,000 to get the real length, or divide a real length by 25\,000 to get the map length. Both lengths must be in the same unit.

The real length comes out in centimetres, which is a huge number. So you usually change it into metres or kilometres. Learn these:

100 \text{ cm} = 1 \text{ m} \qquad 1000 \text{ m} = 1 \text{ km}

So 1 km is 1000 \times 100 = 100\,000 cm. To change centimetres straight into kilometres, divide by 100\,000, not by 1000

A scale in words can be written as a ratio. Put both lengths in centimetres first. For "1 cm represents 3 m", 3 m is 300 cm, so the scale is 1:300

Worked example 3: A ratio scale, map to real life

A canal boat goes from a lock to a bridge, then from the bridge to a marina. On a map with a scale of 1:30\,000, the lock to the bridge is 4.5 cm and the bridge to the marina is 2.5 cm. Work out the real length of the whole trip. Give your answer in kilometres.

  1. Add the two lengths on the map: 4.5 + 2.5 = 7 cm
  2. Map to real life, so multiply: 7 \times 30\,000 = 210\,000 cm

  3. Your turn. Divide by 100 to change 210\,000 cm into metres.

    210\,000 \div 100 = 2100 m

  4. Divide by 1000 to change metres into kilometres: 2100 \div 1000 = 2.1 km
  5. Answer: 2.1 km

Worked example 4: Will it fit on the map?

Ravi is drawing a map with a scale of 1:40\,000. A straight railway line is 9.6 km long. He wants to draw it straight across his page, which is 20 cm wide. Will the railway line fit across the page?

  1. Change the real length into centimetres first, so both lengths are in the same unit.
  2. 9.6 \times 1000 = 9600 m, and 9600 \times 100 = 960\,000 cm
  3. Real life to the map, so divide by 40\,000

  4. Your turn. Work out 960\,000 \div 40\,000

    960\,000 \div 40\,000 = 96 \div 4 = 24 cm

  5. Compare: 24 cm is more than the 20 cm width of the page.
  6. Answer: No. The line would be 24 cm long on the map, which is longer than the 20 cm page.

Scale drawings and plans

A scale drawing is an accurate picture of a real object, such as a room, a garden or a boat, with every length made smaller in the same way. A plan is a scale drawing seen from above.

Use the scale on each length separately. A plan with a scale of 1:50 shows a wall 6 cm long for a real wall of 6 \times 50 = 300 cm, which is 3 m.

Some drawings are on a squared grid, with a sentence such as "each square represents 2 m". Count the squares along the side, then multiply by 2. A side 7 squares long is 7 \times 2 = 14 m in real life.

For a real perimeter or a real area, work out the real lengths first. Then add them for a perimeter, or multiply them for an area. Don't multiply an area on the drawing by the scale.

Worked example 5: The area of a room from its plan

The plan of a rectangular bedroom is drawn to a scale of 1:40. On the plan, the room is 10 cm long and 7.5 cm wide. Work out the real area of the bedroom. Give your answer in m².

  1. Work out the real length and the real width first.
  2. Real length: 10 \times 40 = 400 cm, which is 4 m

  3. Your turn. Work out the real width of the room in metres.

    7.5 \times 40 = 300 cm, which is 3 m

  4. Real area: 4 \times 3 = 12 m²
  5. Answer: 12 m²

Working out a scale

Sometimes you are given one length on the drawing and the same length in real life, and you need the scale. Divide the real length by the length on the drawing.

For example, 4 cm on a map stands for 18 km. Then 18 \div 4 = 4.5, so 1 cm represents 4.5 km.

To write the scale as a ratio 1:n, put both lengths in the same unit first, usually centimetres. Then use the scale on any other length in the drawing.

Worked example 6: Finding a scale, then using it

A model of a tram is 24 cm long. The real tram is 36 m long. (a) Write the scale of the model in the form 1:n (b) On the model, a door is 1.4 cm tall. Work out the height of the real door. Give your answer in metres.

  1. (a) Put both lengths in centimetres: 36 m = 36 \times 100 = 3600 cm
  2. The scale is 24:3600. Divide both sides by 24 to make the first number 1

  3. Your turn. Work out 3600 \div 24

    3600 \div 24 = 150, so the scale is 1:150

  4. (b) Model to real life, so multiply: 1.4 \times 150 = 210 cm
  5. Change to metres: 210 \div 100 = 2.1 m
  6. Answer: (a) 1:150 (b) 2.1 m

Common mistakes

  • Dividing when you should multiply. From the map to real life, multiply by the scale; from real life to the map, divide.
  • Forgetting that a ratio scale is in centimetres. With 1:25\,000, a map length in centimetres gives a real length in centimetres, not metres.
  • Changing centimetres to kilometres by dividing by 1000. There are 100\,000 cm in a kilometre, so divide by 100 and then by 1000
  • Using the scale before the units match. Change the real length into centimetres before you divide by the scale.
  • Giving the answer in the wrong unit, or with no unit. Read which unit the question asks for, change your answer into it and write the unit.
  • Not checking the size of the answer. A real distance should be much bigger than the map distance, and a map length should fit on a page.

Practice questions

More on this topic: Scale Drawings and Maps worksheet with full solutions