Scale drawings and bearings

A scale drawing shows something real at a smaller (or bigger) size. The scale tells you how lengths compare, such as 1 cm to 2 m.

1. Real length = drawing length × what 1 cm stands for.
2. Drawing length = real length ÷ what 1 cm stands for.
3. Angles stay the same.

Example: on a plan, 1 cm represents 2 m. A room is 4.5 cm long on the plan. How long is the real room? How long would a 12 m wall be on the plan?

1. 4.5 × 2 = 9 m
2. 12 ÷ 2 = 6 cm

A map scale such as 1 : 25 000 means 1 cm on the map is 25 000 cm in real life.

1. Map to real: multiply by 25 000 to get cm, then convert to m or km.
2. Real to map: convert to cm first, then divide by 25 000.

Example: two villages are 6.4 cm apart on a 1 : 25 000 map. How far apart are they in real life?

1. 6.4 × 25 000 = 160 000 cm
2. 160 000 ÷ 100 000 = 1.6 km

Example: how long is a 3 km road on the same map?

1. 3 km = 300 000 cm
2. 300 000 ÷ 25 000 = 12 cm

A bearing gives a direction as an angle:

1. measured from north
2. measured clockwise
3. written with three figures, such as 045°.

"The bearing of B from A" means stand at A, face north, and turn clockwise until you face B.

Example: describe how to find the point B on a bearing of 070° from A.

1. Draw a north line at A
2. Measure 70° clockwise from it
3. B is along that line

The bearing of A from B (the back bearing) is the bearing of B from A, plus or minus 180°.

1. If the bearing is less than 180°, add 180°.
2. If it is 180° or more, subtract 180°.

Example: the bearing of B from A is 070°. Find the bearing of A from B.

1. 070° is less than 180°
2. 70 + 180 = 250°

Example: the bearing of Q from P is 300°. Find the bearing of P from Q.

1. 300 − 180 = 120°

This works because the north lines at A and B are parallel, so the angles there are co-interior and add up to 180°.