Scale Diagrams - Standard Maths

What a scale means

A scale tells you how a length on the diagram relates to the real length. It is written either as a ratio, such as \(1 : 25000\), or as a statement, such as 1 cm to 5 m.

Scale: The fixed ratio between a distance on the drawing and the matching real distance. A scale of \(1 : 25000\) means 1 unit on paper stands for 25000 of the same units in real life.

In a ratio scale both sides use the same unit, so \(1 : 25000\) means 1 cm on the map is 25000 cm in reality. You then convert that answer into sensible units such as metres or kilometres.

Converting lengths

There are two directions. To go from the drawing to real life, multiply by the scale. To go from real life to the drawing, divide by the scale.

\[ \text{real length} = \text{drawing length} \times \text{scale} \]

Worked example

On a map with scale \(1 : 25000\), two towns are 8 cm apart. Find the real distance in kilometres.

Multiply the map distance by the scale: \(8 \times 25000 = 200000\) cm.

Convert cm to m by dividing by 100: \(200000 \div 100 = 2000\) m.

Convert m to km by dividing by 1000: \(2000 \div 1000 = 2\) km.

The towns are 2 km apart

Watch the units

A ratio scale gives the real length in the same unit you measured with, usually centimetres. Convert afterwards: divide by 100 for metres, then by 1000 for kilometres. Skipping this step is the classic scale-diagram error.

Bearings

A bearing describes a direction. It is always measured clockwise from north and written with three figures, so an angle of \(60^\circ\) is written as \(060^\circ\).

Bearing: A direction given as a three-figure angle measured clockwise from north, from \(000^\circ\) round to \(360^\circ\). North is \(000^\circ\), east is \(090^\circ\), south is \(180^\circ\) and west is \(270^\circ\).

The bearing of A from B and the bearing of B from A differ by \(180^\circ\). This return direction is called the back bearing: add \(180^\circ\) if the first bearing is under \(180^\circ\), or subtract \(180^\circ\) if it is over.

Worked example

The bearing of a harbour H from a boat B is \(075^\circ\). Find the bearing of the boat B from the harbour H.

The back bearing differs by \(180^\circ\). Since \(075^\circ\) is under \(180^\circ\), add: \(075 + 180\).

Work it out: \(75 + 180 = 255\).

The bearing of B from H is \(255^\circ\)

Reading and drawing

To read a scale diagram, measure the length on paper with a ruler, then convert using the scale. To draw one, reverse it: divide the real length by the scale to get the paper length, then measure and rule it, using a protractor for any bearing.

Worked example

A garden path is 30 m long. It is to be drawn using a scale of 1 cm to 5 m. How long is the path on the drawing?

Here 1 cm stands for 5 m, so divide the real length by 5: \(30 \div 5\).

Work it out: \(30 \div 5 = 6\).

The path is 6 cm long on the drawing

Where this is assessed

Converting cleanly with the scale is Criterion A (Knowing and understanding). A drawing that is accurate to the millimetre and degree, with the scale stated, is Criterion C (Communicating). Planning a route or reading a real map with bearings is Criterion D (Applying maths in context).

Check yourself

1. On a map with scale \(1 : 50000\), a road measures 4 cm. Find the real distance in kilometres. +

Real length \(= 4 \times 50000 = 200000\) cm. Divide by 100 for metres: \(2000\) m, then by 1000 for kilometres: 2 km.

2. A scale drawing uses 1 cm to 2 m. A wall is 15 m long in real life. How long is it on the drawing? +

Divide the real length by the scale: \(15 \div 2 = \mathbf{7.5}\) cm.

3. The bearing of town B from town A is \(075^\circ\). Find the bearing of A from B. +

The back bearing differs by \(180^\circ\). As \(075^\circ\) is under \(180^\circ\), add: \(75 + 180 = \mathbf{255^\circ}\).

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