Bearings & Scale Drawings (Years 8–11) — Maths

A ship's captain, a pilot and a hill walker with a compass all describe direction the same way: with a bearing. And any map you have ever used is a scale drawing — real distances shrunk down so a whole country fits on a page. This chapter covers both, plus the handy trick for finding the bearing back the way you came.

Why this matters: bearings and scale are real navigation maths — used in sailing, aviation, orienteering and surveying. They also sharpen your angle work and your ratio work at the same time, both of which run right through GCSE.

A bearing is measured clockwise from North

Three-figure bearings

Bearings run clockwise from North: N 000°, E 090°, S 180°, W 270°.

A bearing is the angle measured clockwise from North, and it is always written with three figures — think of it like a three-digit combination lock, so 45° becomes 045°.

North = 000° · East = 090° · South = 180° · West = 270°.
An angle of 45° is written 045° — pad with a leading zero. An angle of 9° becomes 009°.

Direction	Bearing
North	000°
East	090°
South	180°
West	270°

Back bearings

The back bearing is a half-turn: 070° out → 070 + 180 = 250° home.

The bearing back the way you came is found with ±180°: add 180 if the bearing is under 180, subtract 180 if it is over. It works because turning to face home is exactly a half-turn (180°) from your outward direction.

Bearing of B from A is 070° → bearing of A from B is 070 + 180 = 250°. And the bearing of 200° gives a back bearing of 200 − 180 = 020°.

Scale drawings

A scale like 1 cm : 5 km means every 1 cm on the drawing is 5 km in real life. To go from drawing to real life you multiply; to go from real life back to the drawing you divide.

On a 1 cm : 5 km map, 4 cm represents 4 × 5 = 20 km. Going the other way, 35 km would need 35 ÷ 5 = 7 cm of drawing.

Scale 1 cm : 5 km — multiply the map length by 5 to reach real distance.

Common mistake: forgetting the leading zero, so 60° gets written as "60°" instead of the correct 060°. On scale questions, the other classic slip is multiplying when you should divide — always check which way you are converting.

In a nutshell: bearings are clockwise from North and always three figures; back bearing = the original ±180°; scale drawings convert by multiplying (to real) or dividing (to drawing) with the given ratio.