Maps, Scale Drawings and Scale Factors — AQA GCSE Maths Revision Notes

What you'll learn

This topic covers representing real distances on paper. By the end of this guide you should be able to read a scale written as a ratio or as a statement, convert a drawing length into a real length and back again, and handle the unit changes that these conversions always involve.

You should also be able to work out a scale from a known pair of distances, use scale drawings together with bearings, and say which of two maps shows more detail.

The organising idea is that a scale is a multiplier, and the only real difficulty is units. Going from the drawing to real life you multiply; going the other way you divide. That part is straightforward. What makes questions hard is that a map scale of 1 : 50 000 is written in the same units on both sides, so it means 1 cm represents 50 000 cm — and the answer is almost always wanted in kilometres. Nearly every lost mark in this topic is a unit conversion, not a multiplication.

Key terms and definitions

Scale — the relationship between a length on a drawing and the corresponding real length.

Ratio scale — a scale written as a ratio, such as 1 : 50 000. Both sides are in the same units.

Statement scale — a scale written in words, such as "1 cm represents 2 km". Here the units differ and are stated.

Scale factor — the number you multiply a drawing length by to get the real length.

Scale drawing — an accurate drawing in which every length is in the same proportion to real life.

Bearing — a direction measured clockwise from north, written with three figures.

Core concepts

Reading a ratio scale

A scale of 1 : n means one unit on the drawing represents n of the same units in reality. The units are not stated because they cancel: 1 cm represents n cm, and equally 1 inch would represent n inches.

So 1 : 100 means 1 cm represents 100 cm, which is 1 m.

And 1 : 50 000 means 1 cm represents 50 000 cm. Converting, that is 500 m, or 0.5 km.

Getting this conversion right at the start makes the rest of the question straightforward, which is why it is worth writing down as a separate line.

The useful conversions

Since ratio scales are almost always given in centimetres and answers wanted in metres or kilometres, two conversions do nearly all the work.

There are 100 cm in a metre and 1000 m in a kilometre, so there are 100 000 cm in a kilometre.

That last figure is the one to remember. A scale of 1 : 100 000 means 1 cm represents exactly 1 km, and every other map scale can be compared against it.

So on a 1 : 25 000 map, 1 cm represents 25 000 cm, which is a quarter of 100 000 cm, so 1 cm represents 0.25 km.

Drawing to real

Multiply the drawing length by the scale factor, then convert the units.

On a 1 : 50 000 map, a distance of 4 cm represents 4 × 50 000 = 200 000 cm. Dividing by 100 000 gives 2 km.

An alternative route is to convert the scale first — 1 cm represents 0.5 km — and then multiply: 4 × 0.5 = 2 km. Both give the same answer, and the second is usually quicker once the scale has been converted.

Real to drawing

Divide the real length by the scale factor, having first put both into the same units.

Two towns 12 km apart on a 1 : 200 000 map: 12 km is 1 200 000 cm, and dividing by 200 000 gives 6 cm.

The direction of the operation is worth checking by sense. A drawing length must be smaller than the real length, so if your answer comes out larger, the multiplication and division have been swapped.

Statement scales

A scale written as "1 cm represents 2 m" has the conversion built into it, which makes it easier to use.

A wall drawn 5 cm long is really 5 × 2 = 10 m. A fence 28 m long is drawn 28 ÷ 4 = 7 cm on a 1 cm : 4 m drawing.

These need no unit conversion at all, provided the answer is left in the units the scale names.

Finding the scale

When a drawing length and the matching real length are both known, convert both to the same units and simplify the ratio.

If 3 cm on a map represents 9 km: 9 km is 900 000 cm, and 3 : 900 000 simplifies by dividing both sides by 3 to give 1 : 300 000.

Always divide until the left-hand side is 1, since that is the standard form for a map scale.

Which map shows more detail?

This catches people out because the comparison runs the opposite way to intuition.

A smaller number after the 1 means a more detailed map. On a 1 : 25 000 map, 1 cm covers only 250 m of ground, so a lot of detail fits on the page. On a 1 : 200 000 map, the same 1 cm covers 2 km, so far less detail can be shown.

Put another way, the larger the second number, the more ground is squeezed into each centimetre.

Models and scale factors

The same arithmetic applies to models. A model car built to 1 : 20 means every real length is 20 times the model's.

A real car 4 m long gives a model of 4 ÷ 20 = 0.2 m, which is 20 cm. Note that the answer was converted to sensible units at the end — 0.2 m is correct but 20 cm is the natural way to describe a model.

Scale drawings with bearings

Locating a point needs both a direction and a distance: the bearing gives the direction, measured clockwise from north as a three-figure number, and the scale converts the distance.

Measure the bearing with a protractor from a north line drawn at the starting point, then measure the distance along that line with a ruler and convert it using the scale.

Worked examples

Example 1: Map to real distance

Two points are 6 cm apart on a 1 : 50 000 map. What is the real distance in kilometres?

First convert the scale: 1 cm represents 50 000 cm. Since there are 100 000 cm in a kilometre, 1 cm represents 0.5 km.

Then multiply: 6 × 0.5 = 3 km.

The alternative route gives the same result: 6 × 50 000 = 300 000 cm, and 300 000 ÷ 100 000 = 3 km.

Example 2: Real distance to map

A field is 250 m by 180 m. What are its dimensions on a 1 : 5000 plan?

Convert the real lengths to centimetres: 250 m is 25 000 cm and 180 m is 18 000 cm.

Divide each by the scale factor: 25 000 ÷ 5000 = 5 cm, and 18 000 ÷ 5000 = 3.6 cm.

The plan measures 5 cm by 3.6 cm.

Both answers are far smaller than the real lengths, which is the sense check that the division was the right way round.

Example 3: Finding a scale

On a drawing, 4 cm represents 2 km. Write the scale in the form 1 : n.

Convert the real distance to the same units: 2 km is 200 000 cm.

The ratio is 4 : 200 000. Divide both sides by 4 to make the left-hand side 1.

The scale is 1 : 50 000.

Common mistakes and how to avoid them

Forgetting that a ratio scale is in the same units on both sides. 1 : 50 000 means 1 cm to 50 000 cm, not to 50 000 m.

Using the wrong conversion. There are 100 000 cm in a kilometre, not 1000.

Multiplying when you should divide. Drawing to real multiplies; real to drawing divides. A drawing length must always come out smaller.

Thinking a bigger scale number means more detail. It is the reverse: 1 : 25 000 shows more detail than 1 : 200 000.

Leaving the answer in centimetres. Convert to the units the question asks for, usually metres or kilometres.

Not simplifying to 1 : n. A map scale is conventionally written with 1 on the left.

Measuring a bearing anticlockwise or from the wrong line. Bearings are clockwise from north and written with three figures.

Exam technique for "Maps, Scale Drawings and Scale Factors"

Convert the scale into a sentence before doing anything else: "1 cm represents 0.5 km". That single line makes every later calculation a one-step multiplication and often earns a method mark.

Write the units at every stage of the working, not just on the final answer. This topic's errors are unit errors, and carrying the units through exposes them.

Sense-check the direction: real lengths are large, drawing lengths are small.

For a scale drawing, use a sharp pencil, a ruler and a protractor, and work to the accuracy the question states — usually within a millimetre and a degree.

Give bearings as three figures, so an angle of 65° is written 065°.

Round sensibly and state the units asked for, converting from centimetres where necessary.

Quick revision summary

A scale is a multiplier. Drawing to real: multiply. Real to drawing: divide. The drawing length is always the smaller one.

A ratio scale uses the same units on both sides, so 1 : 50 000 means 1 cm represents 50 000 cm.

Remember 100 000 cm in a kilometre. So 1 : 100 000 means 1 cm represents exactly 1 km, 1 : 50 000 means 1 cm represents 0.5 km, and 1 : 25 000 means 1 cm represents 0.25 km.

A statement scale such as "1 cm represents 2 m" needs no conversion, provided the answer stays in those units.

To find a scale, convert both lengths to the same units and simplify until the left-hand side is 1.

A smaller second number means a more detailed map: 1 : 25 000 shows more than 1 : 200 000.

For scale drawings with bearings, measure the direction clockwise from north as three figures, then convert the measured distance using the scale.