Scale drawings

What a scale drawing is

A scale drawing is an accurate picture of a real object or place in which every length is multiplied by the same number. Plans of rooms, maps and model kits are all scale drawings. Because every length is scaled by the same factor, the drawing is similar to the real thing: the angles are unchanged, so you can measure angles straight off the drawing.

A scale can be written in two ways:

• In words, such as "1 cm represents 5 m". Each centimetre on the drawing stands for 5 metres in real life.
• As a ratio, such as $1:500$. The real length is 500 times the drawing length, in the same units: 1 cm on the drawing is 500 cm in real life, 1 mm is 500 mm.

The two forms are linked by converting units. "1 cm represents 5 m" means 1 cm represents 500 cm, which is the ratio $1:500$. A ratio scale has no units, so it must always be read in matching units.

Converting with a scale

The unit facts you need: $100\text{cm}=1\text{m}$, and $1000\text{m}=1\text{km}$, so $1\text{km}=100000\text{cm}$.

Drawing to real: multiply. On a map with scale $1:25000$, a distance of 8.4 cm is $8.4\times 25000=210000$ cm in real life. Convert: $210000\text{cm}=2100\text{m}=2.1\text{km}$.

Real to drawing: divide. A real distance of 3.5 km on the same map: first make the units match, $3.5\text{km}=350000\text{cm}$, then $350000÷25000=14$ cm.

A quick way to handle a ratio scale is to work out what 1 cm represents first. For $1:25000$, 1 cm represents $25000$ cm $=250$ m $=0.25$ km. Then 8.4 cm represents $8.4\times 0.25=2.1$ km.

Sense-check every answer. A real distance should be much larger than the drawing distance; a drawing distance of several metres means you multiplied when you should have divided.

Making a scale drawing

To make a scale drawing:

1. Convert every real length to a drawing length using the scale, and write them down before you start.
2. Draw the lines with a ruler (a ruler must be used for all straight edges) and the angles with a protractor. Angles are drawn at their real size.
3. Label the drawing and state the scale.
4. To answer a question, measure the length on your drawing and convert it back to a real length with the scale.

Measured answers are marked with a tolerance, typically 2 mm on the drawing and $2^{°}$ on angles, so accurate drawing matters. Give the measured drawing length in your working, then the real length.

Compass directions

The four main compass points are north (N), east (E), south (S) and west (W), going clockwise. Halfway between them are north-east (NE), south-east (SE), south-west (SW) and north-west (NW).

"Due east" means exactly east, with no north or south in it. If point $D$ is due east of point $C$, then $D$ is on a horizontal line to the right of $C$ on a map with north at the top, and the bearing of $D$ from $C$ is ${090}^{°}$. Similarly, "due north" is exactly north, ${000}^{°}$.

Three-figure bearings

A bearing gives a direction as an angle:

• measured from north,
• turning clockwise,
• written with three figures, from ${000}^{°}$ to ${360}^{°}$.

So north is ${000}^{°}$, east is ${090}^{°}$, south is ${180}^{°}$, west is ${270}^{°}$, north-east is ${045}^{°}$ and south-west is ${225}^{°}$. An angle of $7^{°}$ clockwise from north is written ${007}^{°}$.

The wording tells you where to stand. "The bearing of $B$ from $A$" means: stand at $A$, face north, and turn clockwise until you face $B$. Draw the north line at $A$, the point after the word "from".

To measure a bearing on a drawing, put the protractor's centre on the "from" point with the zero along the north line. If the direction is to the west of the north line, the bearing is more than ${180}^{°}$: measure the angle anticlockwise from north and subtract it from ${360}^{°}$.

Back bearings

If you know the bearing of $B$ from $A$, the bearing of $A$ from $B$ is the opposite direction, which differs by ${180}^{°}$:

• if the bearing is less than ${180}^{°}$, add ${180}^{°}$;
• if it is ${180}^{°}$ or more, subtract ${180}^{°}$.

For the syllabus's own example: if the bearing of $B$ from $A$ is ${025}^{°}$, the bearing of $A$ from $B$ is ${025}^{°}+{180}^{°}={205}^{°}$.

Why it works. The north lines at $A$ and $B$ are parallel. Extend the north line at $B$ downwards, to the south. The angle between south at $B$ and the line $BA$ equals the angle between north at $A$ and $AB$, because they are alternate angles. So from $B$, the direction of $A$ is half a turn (${180}^{°}$) plus that same angle.

Bearings and scale drawings together

Many exam questions combine the two skills: a journey is described by distances and bearings, you make a scale drawing, and then you measure a distance or a bearing from it.

The method:

1. Mark the starting point and draw a north line through it.
2. Measure the first bearing clockwise from north, draw the line, and mark the first distance to scale.
3. Draw a new north line at the new point, parallel to the first, and measure the next bearing from it.
4. Measure what the question asks for, then convert back with the scale.

Before measuring, it often helps to work out angles inside the triangle using bearings. When a turn makes a right angle, Pythagoras gives a check on the measured length.