Set up a grid
Draw north up, east right. Keep a running position (x, y) and the direction you are facing.
Facing | Turn right | Turn left |
|---|---|---|
North | East | West |
East | South | North |
South | West | East |
West | North | South |
A right turn is 90° clockwise; a left turn is 90° anticlockwise.
Final position and distance
Add up the east–west and north–south movements separately. The straight-line distance is √(x² + y²).
Example. 3 km east then 4 km north: √(9 + 16) = 5 km north-east of the start.
Example. 5 km north, turn right and go 3 km, turn right and go 5 km. The north and south legs cancel, leaving 3 km east.
Diagonal directions
North-east, south-east, south-west and north-west lie halfway between the main directions. If A is north of B and C is east of A, C is north-east of B.
Rotations in degrees
Use bearings measured clockwise from north: N = 0°, NE = 45°, E = 90°, SE = 135°, S = 180°, SW = 225°, W = 270°, NW = 315°. Clockwise turns add degrees; anticlockwise turns subtract.
Example. Facing south-east (135°) and turning 135° anticlockwise gives 0°: north.
Shadows
In the morning the sun is in the east, so shadows fall to the west.
In the evening the sun is in the west, so shadows fall to the east.
At noon the shadow is very short.
Example. In the morning, your shadow falls to your right. The shadow points west, and west is on your right when you face south.
Shortcuts and tricks
The standard way is to draw each leg on a grid. These shortcuts give the final position and facing without a full sketch.
Net movement
Add the north–south legs and the east–west legs separately, and only then combine them. Legs in opposite directions cancel.
Right-triangle triplets
Example: 9 km east, then 12 km north.
Standard way: √(9² + 12²) = √225 = 15 km.
Shortcut: 9 : 12 is 3 : 4 scaled by 3, so the distance is 5 × 3 = 15 km. Remember 3-4-5, 5-12-13 and 8-15-17.
Turns in 45° steps
Example: facing north-west, turn 225° clockwise.
Standard way: convert to bearings: north-west is 315°, and 315 + 225 = 540, minus 360 = 180°, which is south.
Shortcut: 225° is five 45° steps clockwise: NW → N → NE → E → SE → S.
Shadows
The sun is in the east in the morning and in the west in the evening, so a morning shadow points west and an evening shadow points east.
Exam tips
Draw every step; don't track turns in your head.
"Turns back" means a 180° turn.
Check whether the question asks for distance, direction, or both.