- In mathematics, scale can have many meanings, but in accurate drawings and constructions scale refers to a
**ratio** - Maps and technical drawings are usually drawn to a scale
- The scale chosen will depend on the area being mapped out
- A map of a classroom will use a large scale and can show a lot more detail
- A map of a country or a world map will use a small scale and can show a much bigger area but in less detail

- Scale can be given as a description, for example “1 cm on the diagram represents 2 m in real life”
- Scale can be given as a ratio, for example “the scale on the map is 1 : 10 000”

- If you are converting from a scale length to a real length, normally you have to multiply
- For example, if the scale is “1 cm on the diagram represents 2 m in real life”
- to convert 6 cm on the diagram to the real-life measurement, we need to multiply by 2 and change the units to m

- So 6 cm = (6 × 2) m = 12 m

- For example, if the scale is “1 cm on the diagram represents 2 m in real life”

- If you are converting from a real length to a scale length, normally you have to divide
- For example, if the scale is “1 cm on the diagram represents 2m in real life”
- to convert 15 m in real life to the length on the diagram, we need to divide by 2 and change the units to cm

- So 15 m = (15 ÷ 2) cm = 7.5 cm

- For example, if the scale is “1 cm on the diagram represents 2m in real life”

- If the scale is not given in unit form (1 :
*n*), then it may help to convert it to unit form first

On a scale drawing of a bridge, 1 cm represents 12 m.

The distance from one side of the bridge to the other is 300 m.

Find this distance on the scale drawing.

Divide the real-life distance by 12

300 cm ÷ 12 = 25

Change the units from m to cm

**25 cm**

- Map scales are normally given as ratios, 1 :
*n*, without any units - For example, Ordnance Survey maps are often 1 : 25 000
- This means 1 cm on the map represents 25 000 cm in real life
- Or more usefully, 1 centimetre on the map represents 250 metres, or 0.25 kilometres, in real life

- The scale can be used to convert lengths found on a map to lengths in real life
- STEP 1

Use a ruler to measure the distance accurately on the map- For example, measuring a length from A to B as 5.8 cm

- STEP 2

Use the scale to find the actual distance in the same units (usually cm)- For example, if the scale is 1 : 150 000, 5.8 × 150 000 = 870 000 cm

- STEP 3

Convert the actual distance to a more suitable unit- For example, 870 000 cm = 8700 m = 8.7 km

- STEP 1

- The scale can also be used to convert lengths found in real life into a length for a map
- STEP 1

Convert the given scale into a scale of 1 cm : [*the units the real-life distance is in*]- For example, if the scale is 1 : 500 000, convert to 1 cm : 5 km

- STEP 2

Divide the real-life distance by the new map scale to find the distance on the map- For example, if the real-life distance is 20 km, 20 ÷ 5 = 4, so the scale length is 4 cm

- STEP 3

Draw this distance on the map if asked to do so

- STEP 1

- Bearings are a measure of direction
- They are
**measured**from the**North** - They are
**measured clockwise** - The angle is always written in
**3 figures**- If the angle is less than 100° the first digit will be a zero

- They are
- If you need to use a map to find a direction, use a protractor to find the angle
- Always begin by drawing the North line
- Measure the angle clockwise from North
- Write the angle using 3 figures

- Knowing the
**compass directions**for the common directions is helpful**Due east**means on a**bearing of 090°**- Draw the line directly to the right

**Due south**means on a**bearing of 180°**- Draw the line vertically downwards

**Due west**means on a**bearing of 270°**- Draw the line directly to the left

**Due north**means on a**bearing of 360° (or 000°)**- Draw the line vertically upwards

- Other directions can be found using these, for example
**Due Northeast**means on a**bearing of 045°**- This is halfway between North (000°) and East (090°)

**Due Southeast**means on a**bearing of 135°****Due Southwest**means on a**bearing of 225°****Due Northwest**means on a**bearing of 315°**

- Using the above bearings for compass directions will help you to estimate angles for other bearings on the map

- If you have many distances to find, or a very small scale, it may help to convert the scale to more suitable units first, just be careful not to mix them up!
- For example, the scale 1 : 500 000 000 can be converted to 1 cm = 5000 km

The map below shows a two peaks in a national park and the scale is 1 : 50 000.

(a)

A drone takes off from Skarface Pike and flies in a straight line to Buttermount. After 4 km, it passes Sickle Tarn.

Mark Sickle Tarn on the map.

Convert the scale to the form 1 cm : … km

1 : 50 000 = 1 cm : 500 m = 1 cm : 0.5 km

To convert 4km to the scale length, using a scale of 1 cm : 0.5 km, divide 4 by 0.5

4 ÷ 0.5 = 8cm

Using a ruler, draw a line from Skarface Pike to Buttermount. Mark the point 8cm away from Skarface Pike on your straight line. This is the location of Sickle Tarn on the map

b)

Skarface Pike is located Southeast of Buttermount. Write the bearing of Buttermount from Skarface Pike.

If Skarface Pike is located Southeast of Buttermount then Buttermount must be located Northwest of Skarface Pike.

The bearing for Northwest is 315°

**315°**

- The steps for converting between scale lengths and real lengths are similar as for maps
- However with scale drawings, the scale may be more likely to be given in the form of a description with units, rather than a ratio

- The scale can be used to convert lengths found on a drawing to a length in real life
- STEP 1

Use a ruler to measure the distance accurately on the scale drawing, if appropriate- For example, measuring a length from A to B as 5.8 cm

- STEP 2

Use the scale to find the actual distance in the appropriate units- For example, using 5.8 cm, if the scale is “1 cm on the drawing represents 1.5 km in real life”, the actual distance = (5.8 × 1.5) km = 8.7 km

- STEP 1
- The scale can also be used to convert lengths found in real life into a length for a scale drawing
- STEP 1

Use the the scale to convert the actual distance to the scale distance- For example, if the scale is “1cm on the drawing is 5 m in real life” and the actual distance is 20 km, the scale distance will be (20 ÷ 5) cm = 4 cm

- STEP 2

Draw this distance on the map if asked to do so

- STEP 1

Below is a scale drawing of a marketplace. 1 cm in the drawing represents 4 m in real life.

Find the distance in real life between *P* and *S*.

Use a ruler to measure the length from *P* to *S*, in cm or in mm

Multiply this length by the given scale

10.7 × 4 = 42.8

Change the units to metres

**42.8 m**

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