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This page sets out the method for scale drawings in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Scale drawings topic page.
When you measure a length on the map and need the real distance, multiply by the scale.
When you know the real length and must draw it, convert to matching units and divide by the scale.
1:50 means multiply the drawing length by 50 to get the real length.
Divide the real length by 50 to get the drawing length.
1:50 with both lengths in centimetres is real cm = drawing cm × 50.
If you want metres, convert after that.
1 cm to 2 km means the scale factor is not 2. It is 2 km per cm.
Convert 2 km into cm before you write 1:n.
Measure the drawing in centimetres, then multiply by the scale.
Do not multiply by the first number of the ratio. That number is 1.
A scale drawing can also carry an angle.
This topic is the length scale. The angle is copied, not multiplied.
A room drawn at 1:50 should come out larger in real life, not smaller.
If the real room is shorter than the drawing, you divided.
Read 1:100 as “1 on the drawing equals 100 in real life”, in the same unit. It does not mean 1cm equals 100m. 100cm is 1m, so 1:100 means 1cm represents 1m. That shortcut is only for 1:100. On a scale of 1:200, 1cm represents 200cm, which is 2m, not 200m.
Check a scale by testing 1 unit. On 1:50, 1cm on the drawing is 50cm in real life, which is 0.5m. On 1:25000, 1cm on the map is 25000cm on the ground. 25000cm=250m=0.25km. If you remember the meaning of one centimetre, every longer line is just that amount multiplied by the number of centimetres you measured.
Method: 1. Read 1:n as 1 unit on the drawing to n of the same unit in real life. 2. On 1:100, 1cm represents 100cm, which is 1m. 3. On 1:200, 1cm represents 200cm, which is 2m.
Measure the drawing in centimetres, multiply by n, and the result is the real length in centimetres. Then change units if the question asks for metres or kilometres. A line of 6cm on a scale of 1:200 is 6× 200=1200cm in real life. Divide by 100 to get metres: 1200÷ 100=12m. Multiplying 6 by 200 gives 1200, and that number is in centimetres. Writing 1200m would be a unit slip. Show the centimetres first.
On a map of scale 1:50000, a distance of 3cm is 3× 50000=150000cm. Convert in stages if that helps: 150000cm=1500m=1.5km. The compact check is 3× 50000÷ 100000=1.5, because there are 100000cm in 1km. Either path is fine if the units are named on each line.
Method: 1. Multiply the drawing length in centimetres by n. 2. A 6cm line on 1:200 is 6× 200=1200cm=12m. 3. A 3cm line on 1:50000 is 3× 50000=150000cm=1.5km.
To find the drawing length, put the real length into the same unit as the drawing, then divide by n. Drawings are almost always measured in centimetres. A real distance of 4m on a scale of 1:100 is 400cm on the ground, and 400÷ 100=4cm on the paper. A wall of 8.5m on a scale of 1:200 is 850cm, then 850÷ 200=4.25cm. Dividing 8.5 by 200 gives 0.0425, which is the answer in metres on the paper, not centimetres. That number will not match a ruler.
The longer the real object, the longer the line, but only after the unit is fixed. Double the real length and you double the drawing. Half the scale number, from 1:200 to 1:100, and the drawing of the same object doubles, because you divide by a smaller n. Students sometimes multiply when they should divide. Ask: is the drawing smaller than the real object? If it is, the drawing number should be smaller, so you divide.
Method: 1. Change the real length into centimetres, then divide by n. 2. A real 4m on 1:100 is 400÷ 100=4cm on the paper. 3. A real 8.5m on 1:200 is 850÷ 200=4.25cm on the paper.
Kilometres, metres and centimetres are the usual chain. 1km=1000m and 1m=100cm, so 1km=1000× 100=100000cm. A real distance of 2.4km is 2.4× 1000=2400m, then 2400× 100=240000cm. On a scale of 1:25000 the drawing is 240000÷ 25000=9.6cm. One line that does the same job is 2.4× 1000× 100÷ 25000=9.6.
Do the unit change and the scale as separate steps so a wrong factor is easy to see. A factor of 1000 moves between kilometres and metres. A factor of 100 moves between metres and centimetres. The scale 25000 is not either of those factors. It belongs only to the 1:25000 statement. Mixing them, for example dividing 2.4 by 25000, skips the conversion and gives a tiny number that is not a length on the page.
Method: 1. Convert kilometres to centimetres before you divide by the scale. 2. Use 1km=100000cm. 3. A real 2.4km on 1:25000 is 2.4× 100000÷ 25000=9.6cm.
A scale can be written two ways. In words, "1 cm represents 5 km" says that one centimetre on the map stands for five kilometres on the ground. As a ratio, 1 : 500 000 says the same thing, because 5 km = 500 000 cm. Exam papers swap between the two forms, so you need to be quick at changing between them.
To change words into a ratio, turn the real distance into the same unit as the drawing unit. For 1 cm representing 5 km: 5 × 1000 × 100 = 500 000, so the ratio is 1 : 500 000. To change a ratio into words, divide by 100 to get metres, and by 1000 again for kilometres. For example, 1 : 25 000 means 1 cm represents 250 m, which is 0.25 km.
Method: 1. Write the scale with the drawing length first, as 1 : n. 2. Change the real distance into centimetres: metres × 100, kilometres × 100 000. 3. To go from the ratio to words, divide by 100 for metres and by 100 000 for kilometres. 4. Check that your answer has the same units as the question asks for.
Tip: Remember the fast facts: 1 m = 100 cm and 1 km = 100 000 cm.
Some exam questions do not give you the scale. Instead they give one length on the drawing and the matching real length, and you must find the ratio. Divide the real length by the drawing length, after changing both to centimetres. If a map distance of 8 cm stands for 2.4 km, then 240 000 ÷ 8 = 30 000, so the scale is 1 : 30 000.
A harder version gives a difference. For example, one road is 2.5 cm longer than another on the map, and in real life it is 1.5 km longer. The difference in map lengths matches the difference in real lengths, so 2.5 cm represents 1.5 km. Divide 150 000 cm by 2.5 to get 60 000. Write the answer as 1 : 60 000 or "1 cm represents 0.6 km".
Method: 1. Write down the drawing length and the real length it stands for. 2. Change the real length to the same unit as the drawing, usually centimetres. 3. Divide the real length by the drawing length to get n in 1 : n. 4. For a difference question, use the difference of the two drawing lengths against the difference of the real lengths.
A scale of 1 : n multiplies every length by n. Areas are made of two lengths, so they are multiplied by n^2, and volumes by n^3. This is why a bedroom drawn as 12 cm by 15 cm on a 1 : 50 plan is 600 cm by 750 cm in real life, an area of 450 000 cm^2, and not 180 × 50.
The safest method is to convert each length first and then find the area. You can also find the area on the drawing and multiply by n^2. Be careful with the final unit. Divide by 10 000 to turn square centimetres into square metres, and by 1 000 000 to turn cubic centimetres into cubic metres. If the question compares two rooms, put both in the same unit before you decide which is bigger.
Method: 1. Multiply each drawing length by n to get the real lengths. 2. Work out the real area (or volume) from the real lengths. 3. Convert to square or cubic metres if the question asks. 4. For a comparison, write both areas in the same unit and give a sentence.
Tip: Area scales by n^2 and volume by n^3. Never multiply an area by n alone.
Exam questions often continue after the distance. A map gives the distance, and the question asks how long a walk takes at a stated speed, or how much a road costs, or how much paint is needed for a plan. Break these into small steps. First scale up to real life, then convert to the unit in the speed or the price, and then carry out the second calculation.
You may also combine scale with bearings. A scale drawing of a boat trip uses a bearing for each leg and a scale for each distance. When you read a measurement from a diagram, accept that it is accurate to about 1 mm and 2^°. Write the final answer in the units the question asks for and round sensibly, since a measured distance rarely deserves more than 2 or 3 significant figures.
Method: 1. Use the scale to find the real distance, then change to metres or kilometres. 2. Use the real distance in the next part: speed, time, cost or area. 3. Round up for items that are bought in whole numbers, such as bags or tins. 4. Check the final answer is a sensible size for the real situation.
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